#set document(title: "8.7 Use Radicals in Functions", author: "OpenStax / XYZ Homework")
#set page(width: 8.5in, height: auto, margin: 1in)
#import "@preview/cetz:0.5.2"
#set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en")
#show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math"))
#set par(justify: true, leading: 0.62em, spacing: 0.9em)
#set enum(spacing: 1.1em)   // room between list items so tall inline fractions don't collide
#set list(spacing: 1.1em)
#set table(stroke: 0.5pt + rgb("#c7ccd3"))

#let BLUE = rgb("#183B6F")     // brand navy — section bars + example/solution labels (white on navy 11.09:1)
#let ORANGE = rgb("#A94509")   // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text)
#let RED = rgb("#DC2626")      // brand error-600
#let GREEN = rgb("#059669")    // brand success-600 (decoration only; small green text uses green-text #007942)

#show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt,
  fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt,
  text(fill: white, weight: "bold", size: 19pt, it.body))
#show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt,
  fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt,
  text(fill: white, weight: "bold", size: 12pt, it.body))
#show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body)
#show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body)

#let examplebox(label, title, body) = block(width: 100%, breakable: true,
  fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt,
  above: 12pt, below: 12pt)[
  #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt,
    text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]]
  #body]

// rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade
#let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint,
    stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt),
    above: 11pt, below: 11pt)[
    #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)]
    #linebreak() #body]

#let solutionbox(body) = block(above: 4pt, below: 8pt)[
  #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body]

#let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"),
  stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[
  #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))]

// Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a
// consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the
// print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a
// degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero.
#let _STD_W = 3.5
#let _WIDE_W = 5.6
#let _MAX_H = 3.4
#let _ASPECT_WIDE = 2.2
#let _UPSCALE_MAX = 1.15
#let stdfig(body) = context {
  let m = measure(body)
  let w = calc.max(m.width / 1in, 0.01)
  let h = calc.max(m.height / 1in, 0.01)
  let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W }
  let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX)
  align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body)))
}

#show figure: set block(breakable: false)
#set figure(gap: 8pt)
#show figure.caption: set text(size: 8.5pt, fill: rgb("#555"))


== 8.7#h(0.6em)Use Radicals in Functions

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Before you get started, take this readiness quiz.

Solve: #math.equation(block: false, alt: "1 minus 2 x greater than or equal to 0 .")[$1 − 2 x ≥ 0 .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "open parenthesis − ∞ , the fraction 1 over 2 ]")[$( "−" ∞ , frac(1, 2) ]$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
For #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 3 x minus 4 ,")[$f ( x ) = 3 x − 4 ,$] evaluate #math.equation(block: false, alt: "f open parenthesis 2 close parenthesis , f open parenthesis −1 close parenthesis , f open parenthesis 0 close parenthesis .")[$f ( 2 ) , f ( −1 ) , f ( 0 ) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "f open parenthesis 2 close parenthesis equals 2 , f open parenthesis −1 close parenthesis equals −7 , f open parenthesis 0 close parenthesis equals −4")[$f ( 2 ) = 2 , f ( −1 ) = −7 , f ( 0 ) = −4$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Graph #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of x .")[$f ( x ) = sqrt(x) .$] State the domain and range of the function in interval notation. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "domain: [ 0 , ∞ close parenthesis ; range: [ 0 , ∞ close parenthesis")[$"domain:" #h(0.5em) [ 0 , ∞ ) ; #h(0.2em) "range:" #h(0.5em) [ 0 , ∞ )$]
]
]


=== Evaluate a Radical Function

In this section we will extend our previous work with functions to include radicals. If a function is defined by a radical expression, we call it a #strong[radical function].

The square root function is #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the root of x .")[$f ( x ) = root(, x) .$]

The cube root function is #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the cube root of x .")[$f ( x ) = root(3, x) .$]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Radical Function]

A #strong[radical function] is a function that is defined by a radical expression.
]

To evaluate a radical function, we find the value of #emph[f] (#emph[x]) for a given value of #emph[x] just as we did in our previous work with functions.

#examplebox("Example 1")[][
For the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of 2 x minus 1 ,")[$f ( x ) = sqrt(2 x − 1) ,$] find ⓐ #math.equation(block: false, alt: "f open parenthesis 5 close parenthesis")[$f ( 5 )$] ⓑ #math.equation(block: false, alt: "f open parenthesis −2 close parenthesis .")[$f ( −2 ) .$]

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of 2 x minus 1")[$f ( x ) = sqrt(2 x − 1)$]]),
  [To evaluate #math.equation(block: false, alt: "f open parenthesis 5 close parenthesis ,")[$f ( 5 ) ,$] substitute 5 for #math.equation(block: false, alt: "x .")[$x .$]], [#math.equation(block: false, alt: "f open parenthesis 5 close parenthesis equals the square root of 2 times 5 minus 1")[$f ( 5 ) = sqrt(2 · 5 − 1)$]],
  [Simplify.], [#math.equation(block: false, alt: "f open parenthesis 5 close parenthesis equals the square root of 9")[$f ( 5 ) = sqrt(9)$]],
  [Take the square root.], [#math.equation(block: false, alt: "f open parenthesis 5 close parenthesis equals 3")[$f ( 5 ) = 3$]],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of 2 x minus 1")[$#h(2em) f ( x ) = sqrt(2 x − 1)$]]),
  [To evaluate #math.equation(block: false, alt: "f open parenthesis −2 close parenthesis ,")[$f ( −2 ) ,$] substitute −2 for #math.equation(block: false, alt: "x .")[$x .$]], [#math.equation(block: false, alt: "f open parenthesis −2 close parenthesis equals the square root of 2 open parenthesis −2 close parenthesis minus 1")[$#h(1.5em) f ( −2 ) = sqrt(2 ( −2 ) − 1)$]],
  [Simplify.], [#math.equation(block: false, alt: "f open parenthesis −2 close parenthesis equals the square root of −5")[$#h(1.5em) f ( −2 ) = sqrt(−5)$]],
))

Since the square root of a negative number is not a real number, the function does not have a value at #math.equation(block: false, alt: "x equals −2 .")[$x = −2 .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
For the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of 3 x minus 2 ,")[$f ( x ) = sqrt(3 x − 2) ,$] find ⓐ #math.equation(block: false, alt: "f open parenthesis 6 close parenthesis")[$f ( 6 )$] ⓑ #math.equation(block: false, alt: "f open parenthesis 0 close parenthesis .")[$f ( 0 ) .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "f open parenthesis 6 close parenthesis equals 4")[$f ( 6 ) = 4$] ⓑ no value at #math.equation(block: false, alt: "x equals 0")[$x = 0$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
For the function #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the square root of 5 x plus 5 ,")[$g ( x ) = sqrt(5 x + 5) ,$] find ⓐ #math.equation(block: false, alt: "g open parenthesis 4 close parenthesis")[$g ( 4 )$] ⓑ #math.equation(block: false, alt: "g open parenthesis −3 close parenthesis .")[$g ( −3 ) .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "g open parenthesis 4 close parenthesis equals 5")[$g ( 4 ) = 5$] ⓑ no value at #math.equation(block: false, alt: "f open parenthesis −3 close parenthesis")[$f ( −3 )$]
]
]

We follow the same procedure to evaluate cube roots.

#examplebox("Example 2")[][
For the function #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the cube root of x minus 6 ,")[$g ( x ) = root(3, x − 6) ,$] find ⓐ #math.equation(block: false, alt: "g open parenthesis 14 close parenthesis")[$g ( 14 )$] ⓑ #math.equation(block: false, alt: "g open parenthesis −2 close parenthesis .")[$g ( −2 ) .$]

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the cube root of x minus 6")[$#h(0.8em) g ( x ) = root(3, x − 6)$]]),
  [To evaluate #math.equation(block: false, alt: "g open parenthesis 14 close parenthesis ,")[$g ( 14 ) ,$] substitute 14 for #math.equation(block: false, alt: "x .")[$x .$]], [#math.equation(block: false, alt: "g open parenthesis 14 close parenthesis equals the cube root of 14 minus 6")[$#h(0.5em) g ( 14 ) = root(3, 14 − 6)$]],
  [Simplify.], [#math.equation(block: false, alt: "g open parenthesis 14 close parenthesis equals the cube root of 8")[$#h(0.5em) g ( 14 ) = root(3, 8)$]],
  [Take the cube root.], [#math.equation(block: false, alt: "g open parenthesis 14 close parenthesis equals 2")[$#h(0.5em) g ( 14 ) = 2$]],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the cube root of x minus 6")[$#h(1em) g ( x ) = root(3, x − 6)$]]),
  [To evaluate #math.equation(block: false, alt: "g open parenthesis −2 close parenthesis ,")[$g ( −2 ) ,$] substitute −2 for #math.equation(block: false, alt: "x .")[$x .$]], [#math.equation(block: false, alt: "g open parenthesis −2 close parenthesis equals the cube root of −2 minus 6")[$#h(0.5em) g ( −2 ) = root(3, −2 − 6)$]],
  [Simplify.], [#math.equation(block: false, alt: "g open parenthesis −2 close parenthesis equals the cube root of −8")[$#h(0.5em) g ( −2 ) = root(3, −8)$]],
  [Take the cube root.], [#math.equation(block: false, alt: "g open parenthesis −2 close parenthesis equals −2")[$#h(0.5em) g ( −2 ) = −2$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
For the function #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the cube root of 3 x minus 4 ,")[$g ( x ) = root(3, 3 x − 4) ,$] find ⓐ #math.equation(block: false, alt: "g open parenthesis 4 close parenthesis")[$g ( 4 )$] ⓑ #math.equation(block: false, alt: "g open parenthesis 1 close parenthesis .")[$g ( 1 ) .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "g open parenthesis 4 close parenthesis equals 2")[$g ( 4 ) = 2$] ⓑ #math.equation(block: false, alt: "g open parenthesis 1 close parenthesis equals −1")[$g ( 1 ) = −1$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
For the function #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals the cube root of 5 x minus 2 ,")[$h ( x ) = root(3, 5 x − 2) ,$] find ⓐ #math.equation(block: false, alt: "h open parenthesis 2 close parenthesis")[$h ( 2 )$] ⓑ #math.equation(block: false, alt: "h open parenthesis −5 close parenthesis .")[$h ( −5 ) .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "h open parenthesis 2 close parenthesis equals 2")[$h ( 2 ) = 2$] #linebreak() ⓑ #math.equation(block: false, alt: "h open parenthesis −5 close parenthesis equals −3")[$h ( −5 ) = −3$]
]
]

The next example has fourth roots.

#examplebox("Example 3")[][
For the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the 4th root of 5 x minus 4 ,")[$f ( x ) = root(4, 5 x − 4) ,$] find ⓐ #math.equation(block: false, alt: "f open parenthesis 4 close parenthesis")[$f ( 4 )$] ⓑ #math.equation(block: false, alt: "f open parenthesis −12 close parenthesis")[$f ( −12 )$]

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the 4th root of 5 x minus 4")[$#h(3.5em) f ( x ) = root(4, 5 x − 4)$]]),
  [To evaluate #math.equation(block: false, alt: "f open parenthesis 4 close parenthesis ,")[$f ( 4 ) ,$] substitute 4 for #math.equation(block: false, alt: "x .")[$x .$]], [#math.equation(block: false, alt: "f open parenthesis 4 close parenthesis equals the 4th root of 5 times 4 minus 4")[$#h(3.5em) f ( 4 ) = root(4, 5 · 4 − 4)$]],
  [Simplify.], [#math.equation(block: false, alt: "f open parenthesis 4 close parenthesis equals the 4th root of 16")[$#h(3.5em) f ( 4 ) = root(4, 16)$]],
  [Take the fourth root.], [#math.equation(block: false, alt: "f open parenthesis 4 close parenthesis equals 2")[$#h(3.5em) f ( 4 ) = 2$]],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the 4th root of 5 x minus 4")[$#h(4.5em) f ( x ) = root(4, 5 x − 4)$]]),
  [To evaluate #math.equation(block: false, alt: "f open parenthesis −12 close parenthesis ,")[$f ( −12 ) ,$] substitute −12 for #math.equation(block: false, alt: "x .")[$x .$]], [#math.equation(block: false, alt: "f open parenthesis −12 close parenthesis equals the 4th root of 5 open parenthesis −12 close parenthesis minus 4")[$#h(3.5em) f ( −12 ) = root(4, 5 ( −12 ) − 4)$]],
  [Simplify.], [#math.equation(block: false, alt: "f open parenthesis −12 close parenthesis equals the 4th root of −64")[$#h(3.5em) f ( −12 ) = root(4, −64)$]],
))

Since the fourth root of a negative number is not a real number, the function does not have a value at #math.equation(block: false, alt: "x equals −12 .")[$x = −12 .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
For the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the 4th root of 3 x plus 4 ,")[$f ( x ) = root(4, 3 x + 4) ,$] find ⓐ #math.equation(block: false, alt: "f open parenthesis 4 close parenthesis")[$f ( 4 )$] ⓑ #math.equation(block: false, alt: "f open parenthesis −1 close parenthesis .")[$f ( −1 ) .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "f open parenthesis 4 close parenthesis equals 2")[$f ( 4 ) = 2$] ⓑ #math.equation(block: false, alt: "f open parenthesis −1 close parenthesis equals 1")[$f ( −1 ) = 1$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
For the function #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the 4th root of 5 x plus 1 ,")[$g ( x ) = root(4, 5 x + 1) ,$] find ⓐ #math.equation(block: false, alt: "g open parenthesis 16 close parenthesis")[$g ( 16 )$] ⓑ #math.equation(block: false, alt: "g open parenthesis 3 close parenthesis .")[$g ( 3 ) .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "g open parenthesis 16 close parenthesis equals 3")[$g ( 16 ) = 3$] ⓑ #math.equation(block: false, alt: "g open parenthesis 3 close parenthesis equals 2")[$g ( 3 ) = 2$]
]
]


=== Find the Domain of a Radical Function

To find the #strong[domain] and #strong[range] of radical functions, we use our properties of radicals. For a radical with an even index, we said the radicand had to be greater than or equal to zero as even roots of negative numbers are not real numbers. For an odd index, the radicand can be any real number. We restate the properties here for reference.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Properties of #math.equation(block: false, alt: "the nth root of a")[$root(n, a)$]]

When #emph[n] is an #strong[even] number and:

- #math.equation(block: false, alt: "a greater than or equal to 0 ,")[$a ≥ 0 ,$] then #math.equation(block: false, alt: "the nth root of a")[$root(n, a)$] is a real number.
- #math.equation(block: false, alt: "a less than 0 ,")[$a < 0 ,$] then #math.equation(block: false, alt: "the nth root of a")[$root(n, a)$] is not a real number.

When #emph[n] is an #strong[odd] number, #math.equation(block: false, alt: "the nth root of a")[$root(n, a)$] is a real number for all values of #emph[a].
]

So, to find the domain of a radical function with even index, we set the radicand to be greater than or equal to zero. For an odd index radical, the radicand can be any real number.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Domain of a Radical Function]

When the #strong[index] of the radical is #strong[even], the radicand must be greater than or equal to zero.

When the #strong[index] of the radical is #strong[odd], the radicand can be any real number.
]

#examplebox("Example 4")[][
Find the domain of the function, #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of 3 x minus 4 .")[$f ( x ) = sqrt(3 x − 4) .$] Write the domain in interval notation.

#solutionbox[
Since the function, #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of 3 x minus 4")[$f ( x ) = sqrt(3 x − 4)$] has a radical with an index of 2, which is even, we know the radicand must be greater than or equal to 0. We set the radicand to be greater than or equal to 0 and then solve to find the domain.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "3 x minus 4 greater than or equal to 0")[$3 x − 4 ≥ 0$]]),
  [Solve.], [#math.equation(block: false, alt: "3 x greater than or equal to 4")[$3 x ≥ 4$]],
  [], [#math.equation(block: false, alt: "x greater than or equal to the fraction 4 over 3")[$x ≥ frac(4, 3)$]],
))

The domain of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of 3 x minus 4")[$f ( x ) = sqrt(3 x − 4)$] is all values #math.equation(block: false, alt: "x greater than or equal to the fraction 4 over 3")[$x ≥ frac(4, 3)$] and we write it in interval notation as #math.equation(block: false, alt: "[ the fraction 4 over 3 , ∞ close parenthesis .")[$[ frac(4, 3) , ∞ ) .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the domain of the function, #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of 6 x minus 5 .")[$f ( x ) = sqrt(6 x − 5) .$] Write the domain in interval notation.

#solutionbox[
#math.equation(block: true, alt: "[ the fraction 5 over 6 , ∞ close parenthesis")[$[ frac(5, 6) , ∞ )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the domain of the function, #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of 4 minus 5 x .")[$f ( x ) = sqrt(4 − 5 x) .$] Write the domain in interval notation.

#solutionbox[
#math.equation(block: true, alt: "open parenthesis − ∞ , the fraction 4 over 5 ]")[$( "−" ∞ , frac(4, 5) ]$]
]
]

#examplebox("Example 5")[][
Find the domain of the function, #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the square root of the fraction 6 over x minus 1 .")[$g ( x ) = sqrt(frac(6, x − 1)) .$] Write the domain in interval notation.

#solutionbox[
Since the function, #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the square root of the fraction 6 over x minus 1")[$g ( x ) = sqrt(frac(6, x − 1))$] has a radical with an index of 2, which is even, we know the radicand must be greater than or equal to 0.

The radicand cannot be zero since the numerator is not zero.

For #math.equation(block: false, alt: "the fraction 6 over x minus 1")[$frac(6, x − 1)$] to be greater than zero, the denominator must be positive since the numerator is positive. We know a positive divided by a positive is positive.

We set #math.equation(block: false, alt: "x minus 1 greater than 0")[$x − 1 > 0$] and solve.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  [#linebreak() Solve.], [#math.equation(block: false, alt: "x minus 1, greater than, 0; x, greater than, 1")[$x − 1 & > & 0 \ x & > & 1$]],
))

Also, since the radicand is a fraction, we must realize that the denominator cannot be zero.

We solve #math.equation(block: false, alt: "x minus 1 equals 0")[$x − 1 = 0$] to find the value that must be eliminated from the domain.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  [#linebreak() Solve.], [#math.equation(block: false, alt: "x minus 1, equals, 0; x, equals, 1 so x not equal to 1 in the domain.")[$x − 1 & = & 0 \ x & = & 1 #h(0.2em) "so" #h(0.2em) x ≠ 1 #h(0.2em) "in the domain."$]],
))

Putting this together we get the domain is #math.equation(block: false, alt: "x greater than 1")[$x > 1$] and we write it as #math.equation(block: false, alt: "open parenthesis 1 , ∞ close parenthesis .")[$( 1 , ∞ ) .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the domain of the function, #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of the fraction 4 over x plus 3 .")[$f ( x ) = sqrt(frac(4, x + 3)) .$] Write the domain in interval notation.

#solutionbox[
#math.equation(block: true, alt: "open parenthesis −3 , ∞ close parenthesis")[$( −3 , ∞ )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the domain of the function, #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals the square root of the fraction 9 over x minus 5 .")[$h ( x ) = sqrt(frac(9, x − 5)) .$] Write the domain in interval notation.

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 5 , ∞ close parenthesis")[$( 5 , ∞ )$]
]
]

The next example involves a cube root and so will require different thinking.

#examplebox("Example 6")[][
Find the domain of the function, #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the cube root of 2 x squared plus 3 .")[$f ( x ) = root(3, 2 x^(2) + 3) .$] Write the domain in interval notation.

#solutionbox[
Since the function, #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the cube root of 2 x squared plus 3")[$f ( x ) = root(3, 2 x^(2) + 3)$] has a radical with an index of 3, which is odd, we know the radicand can be any real number. This tells us the domain is any real number. In interval notation, we write #math.equation(block: false, alt: "open parenthesis − ∞ , ∞ close parenthesis .")[$( "−" ∞ , ∞ ) .$]

The domain of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the cube root of 2 x squared plus 3")[$f ( x ) = root(3, 2 x^(2) + 3)$] is all real numbers and we write it in interval notation as #math.equation(block: false, alt: "open parenthesis − ∞ , ∞ close parenthesis .")[$( "−" ∞ , ∞ ) .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the domain of the function, #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the cube root of 3 x squared minus 1 .")[$f ( x ) = root(3, 3 x^(2) − 1) .$] Write the domain in interval notation.

#solutionbox[
#math.equation(block: true, alt: "open parenthesis − ∞ , ∞ close parenthesis")[$( "−" ∞ , ∞ )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the domain of the function, #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the cube root of 5 x minus 4 .")[$g ( x ) = root(3, 5 x − 4) .$] Write the domain in interval notation.

#solutionbox[
#math.equation(block: true, alt: "open parenthesis − ∞ , ∞ close parenthesis")[$( "−" ∞ , ∞ )$]
]
]


=== Graph Radical Functions

Before we graph any radical function, we first find the #strong[domain] of the function. For the function, #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of x ,")[$f ( x ) = sqrt(x) ,$] the index is even, and so the radicand must be greater than or equal to 0.

This tells us the domain is #math.equation(block: false, alt: "x greater than or equal to 0")[$x ≥ 0$] and we write this in interval notation as #math.equation(block: false, alt: "[ 0 , ∞ close parenthesis .")[$[ 0 , ∞ ) .$]

Previously we used point plotting to graph the function, #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of x .")[$f ( x ) = sqrt(x) .$] We chose #emph[x]-values, substituted them in and then created a chart. Notice we chose points that are perfect squares in order to make taking the square root easier.

#figure(figph[The figure shows the square root function graph on the x y-coordinate plane. The x-axis of the plane runs from 0 to 7. The y-axis runs from 0 to 7. The function has a starting point at (0, 0) and goes through the points (1, 1) and (4, 2). A table is shown beside the graph with 3 columns and 5 rows. The first row is a header row with the expressions “x”, “f (x) = square root of x”, and “(x, f (x))”. The second row has the numbers 0, 0, and (0, 0). The third row has the numbers 1, 1, and (1, 1). The fourth row has the numbers 4, 2, and (4, 2). The fifth row has the numbers 9, 3, and (9, 3).], alt: "The figure shows the square root function graph on the x y-coordinate plane. The x-axis of the plane runs from 0 to 7. The y-axis runs from 0 to 7. The function has a starting point at (0, 0) and goes through the points (1, 1) and (4, 2). A table is shown beside the graph with 3 columns and 5 rows. The first row is a header row with the expressions “x”, “f (x) = square root of x”, and “(x, f (x))”. The second row has the numbers 0, 0, and (0, 0). The third row has the numbers 1, 1, and (1, 1). The fourth row has the numbers 4, 2, and (4, 2). The fifth row has the numbers 9, 3, and (9, 3).", caption: none)

Once we see the graph, we can find the range of the function. The #emph[y]-values of the function are greater than or equal to zero. The range then is #math.equation(block: false, alt: "[ 0 , ∞ close parenthesis .")[$[ 0 , ∞ ) .$]

#examplebox("Example 7")[][
For the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of x plus 3 ,")[$f ( x ) = sqrt(x + 3) ,$]

ⓐ find the domain  ⓑ graph the function ⓒ use the graph to determine the range.

#solutionbox[
ⓐ Since the radical has index 2, we know the radicand must be greater than or equal to zero. If #math.equation(block: false, alt: "x plus 3 greater than or equal to 0 ,")[$x + 3 ≥ 0 ,$] then #math.equation(block: false, alt: "x greater than or equal to −3 .")[$x ≥ −3 .$] This tells us the domain is all values #math.equation(block: false, alt: "x greater than or equal to −3")[$x ≥ −3$] and written in interval notation as #math.equation(block: false, alt: "[ −3 , ∞ close parenthesis .")[$[ −3 , ∞ ) .$]

ⓑ To graph the function, we choose points in the interval #math.equation(block: false, alt: "[ −3 , ∞ close parenthesis")[$[ −3 , ∞ )$] that will also give us a radicand which will be easy to take the square root.

#figure(figph[The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 3 to 3. The y-axis runs from 0 to 7. The function has a starting point at (negative 3, 0) and goes through the points (negative 2, 1) and (1, 2). A table is shown beside the graph with 3 columns and 5 rows. The first row is a header row with the expressions “x”, “f (x) = square root of the quantity x plus 3”, and “(x, f (x))”. The second row has the numbers negative 3, 0, and (negative 3, 0). The third row has the numbers negative 2, 1, and (negative 2, 1). The fourth row has the numbers 1, 2, and (1, 2). The fifth row has the numbers 6, 3, and (6, 3).], alt: "The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 3 to 3. The y-axis runs from 0 to 7. The function has a starting point at (negative 3, 0) and goes through the points (negative 2, 1) and (1, 2). A table is shown beside the graph with 3 columns and 5 rows. The first row is a header row with the expressions “x”, “f (x) = square root of the quantity x plus 3”, and “(x, f (x))”. The second row has the numbers negative 3, 0, and (negative 3, 0). The third row has the numbers negative 2, 1, and (negative 2, 1). The fourth row has the numbers 1, 2, and (1, 2). The fifth row has the numbers 6, 3, and (6, 3).", caption: none)

ⓒ Looking at the graph, we see the #emph[y]-values of the function are greater than or equal to zero. The range then is #math.equation(block: false, alt: "[ 0 , ∞ close parenthesis .")[$[ 0 , ∞ ) .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
For the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of x plus 2 ,")[$f ( x ) = sqrt(x + 2) ,$] ⓐ find the domain  ⓑ graph the function ⓒ use the graph to determine the range.

#solutionbox[
ⓐ domain: #math.equation(block: false, alt: "[ −2 , ∞ close parenthesis")[$[ −2 , ∞ )$] #linebreak() 
ⓑ  #linebreak() 
#figure(figph[The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 2 to 6. The y-axis runs from 0 to 8. The function has a starting point at (negative 2, 0) and goes through the points (negative 1, 1) and (2, 2).], alt: "The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 2 to 6. The y-axis runs from 0 to 8. The function has a starting point at (negative 2, 0) and goes through the points (negative 1, 1) and (2, 2).", caption: none)

 #linebreak() 
ⓒ range: #math.equation(block: false, alt: "[ 0 , ∞ close parenthesis")[$[ 0 , ∞ )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
For the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of x minus 2 ,")[$f ( x ) = sqrt(x − 2) ,$] ⓐ find the domain  ⓑ graph the function ⓒ use the graph to determine the range.

#solutionbox[
ⓐ domain: #math.equation(block: false, alt: "[ 2 , ∞ close parenthesis")[$[ 2 , ∞ )$] #linebreak() 
ⓑ  #linebreak() 
#figure(figph[The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from 0 to 8. The y-axis runs from 0 to 6. The function has a starting point at (2, 0) and goes through the points (3, 1) and (6, 2).], alt: "The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from 0 to 8. The y-axis runs from 0 to 6. The function has a starting point at (2, 0) and goes through the points (3, 1) and (6, 2).", caption: none)

 #linebreak() 
ⓒ range: #math.equation(block: false, alt: "[ 0 , ∞ close parenthesis")[$[ 0 , ∞ )$]
]
]

In our previous work graphing functions, we graphed #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x cubed")[$f ( x ) = x^(3)$] but we did not graph the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the cube root of x .")[$f ( x ) = root(3, x) .$] We will do this now in the next example.

#examplebox("Example 8")[][
For the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the cube root of x ,")[$f ( x ) = root(3, x) ,$] ⓐ find the domain  ⓑ graph the function ⓒ use the graph to determine the range.

#solutionbox[
ⓐ Since the radical has index 3, we know the radicand can be any real number. This tells us the domain is all real numbers and written in interval notation as #math.equation(block: false, alt: "open parenthesis − ∞ , ∞ close parenthesis")[$( "−" ∞ , ∞ )$]

ⓑ To graph the function, we choose points in the interval #math.equation(block: false, alt: "open parenthesis − ∞ , ∞ close parenthesis")[$( "−" ∞ , ∞ )$] that will also give us a radicand which will be easy to take the cube root.

#figure(figph[The figure shows the cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The function has a center point at (0, 0) and goes through the points (1, 1), (negative 1, negative 1), (8, 2), and (negative 8, negative 2). A table is shown beside the graph with 3 columns and 6 rows. The first row is a header row with the expressions “x”, “f (x) = cube root of x”, and “(x, f (x))”. The second row has the numbers negative 8, negative 2, and (negative 8, negative 2). The third row has the numbers negative 1, negative 1, and (negative 1, negative 1). The fourth row has the numbers 0, 0, and (0, 0). The fifth row has the numbers 1, 1, and (1, 1). The sixth row has the numbers 8, 2, and (8, 2).], alt: "The figure shows the cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The function has a center point at (0, 0) and goes through the points (1, 1), (negative 1, negative 1), (8, 2), and (negative 8, negative 2). A table is shown beside the graph with 3 columns and 6 rows. The first row is a header row with the expressions “x”, “f (x) = cube root of x”, and “(x, f (x))”. The second row has the numbers negative 8, negative 2, and (negative 8, negative 2). The third row has the numbers negative 1, negative 1, and (negative 1, negative 1). The fourth row has the numbers 0, 0, and (0, 0). The fifth row has the numbers 1, 1, and (1, 1). The sixth row has the numbers 8, 2, and (8, 2).", caption: none)

ⓒ Looking at the graph, we see the #emph[y]-values of the function are all real numbers. The range then is #math.equation(block: false, alt: "open parenthesis − ∞ , ∞ close parenthesis .")[$( "−" ∞ , ∞ ) .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
For the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals − the cube root of x ,")[$f ( x ) = "−" root(3, x) ,$]

ⓐ find the domain  ⓑ graph the function ⓒ use the graph to determine the range.

#solutionbox[
ⓐ domain: #math.equation(block: false, alt: "( minus ∞ , ∞ close parenthesis")[$"(" − ∞ , ∞ \)$] #linebreak() 
ⓑ  #linebreak() 
#figure(figph[The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 2 to 2. The y-axis runs from negative 2 to 2. The function has a center point at (0, 0) and goes through the points (1, negative 1) and (negative 1, 1).], alt: "The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 2 to 2. The y-axis runs from negative 2 to 2. The function has a center point at (0, 0) and goes through the points (1, negative 1) and (negative 1, 1).", caption: none)

 #linebreak() 
ⓒ range: #math.equation(block: false, alt: "( minus ∞ , ∞ close parenthesis")[$"(" − ∞ , ∞ \)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
For the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the cube root of x minus 2 ,")[$f ( x ) = root(3, x − 2) ,$]

ⓐ find the domain ⓑ graph the function ⓒ use the graph to determine the range.

#solutionbox[
ⓐ domain: #math.equation(block: false, alt: "( minus ∞ , ∞ close parenthesis")[$"(" − ∞ , ∞ \)$] #linebreak() 
ⓑ  #linebreak() 
#figure(figph[The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 1 to 5. The y-axis runs from negative 3 to 3. The function has a center point at (2, 0) and goes through the points (1, negative 1) and (3, 2).], alt: "The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 1 to 5. The y-axis runs from negative 3 to 3. The function has a center point at (2, 0) and goes through the points (1, negative 1) and (3, 2).", caption: none)

 #linebreak() 
ⓒ range: #math.equation(block: false, alt: "( minus ∞ , ∞ close parenthesis")[$"(" − ∞ , ∞ \)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Access these online resources for additional instruction and practice with radical functions.

- #link("https://openstax.org/l/37RadFuncDom1")[Domain of a Radical Function]
- #link("https://openstax.org/l/37RadFuncDom2")[Domain of a Radical Function 2]
- #link("https://openstax.org/l/37RadFuncDom3")[Finding Domain of a Radical Function]
]


=== Key Concepts

- #strong[Properties of #math.equation(block: false, alt: "the nth root of a")[$root(n, a)$]]
- When #emph[n] is an #strong[even] number and: #linebreak() 
#math.equation(block: false, alt: "a greater than or equal to 0 ,")[$a ≥ 0 ,$] then #math.equation(block: false, alt: "the nth root of a")[$root(n, a)$] is a real number. #linebreak() 
#math.equation(block: false, alt: "a less than 0 ,")[$a < 0 ,$] then #math.equation(block: false, alt: "the nth root of a")[$root(n, a)$] is not a real number.
- When #emph[n] is an #strong[odd] number, #math.equation(block: false, alt: "the nth root of a")[$root(n, a)$] is a real number for all values of #emph[a].
- #strong[Domain of a Radical Function]
- When the #strong[index] of the radical is #strong[even], the radicand must be greater than or equal to zero.
- When the #strong[index] of the radical is #strong[odd], the radicand can be any real number.


==== Practice Makes Perfect

#strong[Evaluate a Radical Function]

In the following exercises, evaluate each function.

#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of 4 x minus 4 ,")[$f ( x ) = sqrt(4 x − 4) ,$] find ⓐ #math.equation(block: false, alt: "f open parenthesis 5 close parenthesis")[$f ( 5 )$] ⓑ #math.equation(block: false, alt: "f open parenthesis 0 close parenthesis .")[$f ( 0 ) .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "f open parenthesis 5 close parenthesis equals 4")[$f ( 5 ) = 4$] ⓑ no value at #math.equation(block: false, alt: "x equals 0")[$x = 0$]
]

#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of 6 x minus 5 ,")[$f ( x ) = sqrt(6 x − 5) ,$] find ⓐ #math.equation(block: false, alt: "f open parenthesis 5 close parenthesis")[$f ( 5 )$] ⓑ #math.equation(block: false, alt: "f open parenthesis −1 close parenthesis .")[$f ( −1 ) .$]

#math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the square root of 6 x plus 1 ,")[$g ( x ) = sqrt(6 x + 1) ,$] find ⓐ #math.equation(block: false, alt: "g open parenthesis 4 close parenthesis")[$g ( 4 )$] ⓑ #math.equation(block: false, alt: "g open parenthesis 8 close parenthesis .")[$g ( 8 ) .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "g open parenthesis 4 close parenthesis equals 5")[$g ( 4 ) = 5$] ⓑ #math.equation(block: false, alt: "g open parenthesis 8 close parenthesis equals 7")[$g ( 8 ) = 7$]
]

#math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the square root of 3 x plus 1 ,")[$g ( x ) = sqrt(3 x + 1) ,$] find ⓐ #math.equation(block: false, alt: "g open parenthesis 8 close parenthesis")[$g ( 8 )$] ⓑ #math.equation(block: false, alt: "g open parenthesis 5 close parenthesis .")[$g ( 5 ) .$]

#math.equation(block: false, alt: "F open parenthesis x close parenthesis equals the square root of 3 minus 2 x ,")[$F ( x ) = sqrt(3 − 2 x) ,$] find ⓐ #math.equation(block: false, alt: "F open parenthesis 1 close parenthesis")[$F ( 1 )$] ⓑ #math.equation(block: false, alt: "F open parenthesis −11 close parenthesis .")[$F ( −11 ) .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "F open parenthesis 1 close parenthesis equals 1")[$F ( 1 ) = 1$] ⓑ #math.equation(block: false, alt: "F open parenthesis −11 close parenthesis equals 5")[$F ( −11 ) = 5$]
]

#math.equation(block: false, alt: "F open parenthesis x close parenthesis equals the square root of 8 minus 4 x ,")[$F ( x ) = sqrt(8 − 4 x) ,$] find ⓐ #math.equation(block: false, alt: "F open parenthesis 1 close parenthesis")[$F ( 1 )$] ⓑ #math.equation(block: false, alt: "F open parenthesis −2 close parenthesis .")[$F ( −2 ) .$]

#math.equation(block: false, alt: "G open parenthesis x close parenthesis equals the square root of 5 x minus 1 ,")[$G ( x ) = sqrt(5 x − 1) ,$] find ⓐ #math.equation(block: false, alt: "G open parenthesis 5 close parenthesis")[$G ( 5 )$] ⓑ #math.equation(block: false, alt: "G open parenthesis 2 close parenthesis .")[$G ( 2 ) .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "G open parenthesis 5 close parenthesis equals 2 the square root of 6")[$G ( 5 ) = 2 sqrt(6)$] ⓑ #math.equation(block: false, alt: "G open parenthesis 2 close parenthesis equals 3")[$G ( 2 ) = 3$]
]

#math.equation(block: false, alt: "G open parenthesis x close parenthesis equals the square root of 4 x plus 1 ,")[$G ( x ) = sqrt(4 x + 1) ,$] find ⓐ #math.equation(block: false, alt: "G open parenthesis 11 close parenthesis")[$G ( 11 )$] ⓑ #math.equation(block: false, alt: "G open parenthesis 2 close parenthesis .")[$G ( 2 ) .$]

#math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the cube root of 2 x minus 4 ,")[$g ( x ) = root(3, 2 x − 4) ,$] find ⓐ #math.equation(block: false, alt: "g open parenthesis 6 close parenthesis")[$g ( 6 )$] ⓑ #math.equation(block: false, alt: "g open parenthesis −2 close parenthesis .")[$g ( −2 ) .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "g open parenthesis 6 close parenthesis equals 2")[$g ( 6 ) = 2$] ⓑ #math.equation(block: false, alt: "g open parenthesis −2 close parenthesis equals −2")[$g ( −2 ) = −2$]
]

#math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the cube root of 7 x minus 1 ,")[$g ( x ) = root(3, 7 x − 1) ,$] find ⓐ #math.equation(block: false, alt: "g open parenthesis 4 close parenthesis")[$g ( 4 )$] ⓑ #math.equation(block: false, alt: "g open parenthesis −1 close parenthesis .")[$g ( −1 ) .$]

#math.equation(block: false, alt: "h open parenthesis x close parenthesis equals the cube root of x squared minus 4 ,")[$h ( x ) = root(3, x^(2) − 4) ,$] find ⓐ #math.equation(block: false, alt: "h open parenthesis −2 close parenthesis")[$h ( −2 )$] ⓑ #math.equation(block: false, alt: "h open parenthesis 6 close parenthesis .")[$h ( 6 ) .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "h open parenthesis −2 close parenthesis equals 0")[$h ( −2 ) = 0$] ⓑ #math.equation(block: false, alt: "h open parenthesis 6 close parenthesis equals 2 the cube root of 4")[$h ( 6 ) = 2 root(3, 4)$]
]

#math.equation(block: false, alt: "h open parenthesis x close parenthesis equals the cube root of x squared plus 4 ,")[$h ( x ) = root(3, x^(2) + 4) ,$] find ⓐ #math.equation(block: false, alt: "h open parenthesis −2 close parenthesis")[$h ( −2 )$] ⓑ #math.equation(block: false, alt: "h open parenthesis 6 close parenthesis .")[$h ( 6 ) .$]

For the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the 4th root of 2 x cubed ,")[$f ( x ) = root(4, 2 x^(3)) ,$] find ⓐ #math.equation(block: false, alt: "f open parenthesis 0 close parenthesis")[$f ( 0 )$] ⓑ #math.equation(block: false, alt: "f open parenthesis 2 close parenthesis .")[$f ( 2 ) .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "f open parenthesis 0 close parenthesis equals 0")[$f ( 0 ) = 0$] ⓑ #math.equation(block: false, alt: "f open parenthesis 2 close parenthesis equals 2")[$f ( 2 ) = 2$]
]

For the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the 4th root of 3 x cubed ,")[$f ( x ) = root(4, 3 x^(3)) ,$] find ⓐ #math.equation(block: false, alt: "f open parenthesis 0 close parenthesis")[$f ( 0 )$] ⓑ #math.equation(block: false, alt: "f open parenthesis 3 close parenthesis .")[$f ( 3 ) .$]

For the function #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the 4th root of 4 minus 4 x ,")[$g ( x ) = root(4, 4 − 4 x) ,$] find ⓐ #math.equation(block: false, alt: "g open parenthesis 1 close parenthesis")[$g ( 1 )$] ⓑ #math.equation(block: false, alt: "g open parenthesis −3 close parenthesis .")[$g ( −3 ) .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "g open parenthesis 1 close parenthesis equals 0")[$g ( 1 ) = 0$] ⓑ #math.equation(block: false, alt: "g open parenthesis −3 close parenthesis equals 2")[$g ( −3 ) = 2$]
]

For the function #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the 4th root of 8 minus 4 x ,")[$g ( x ) = root(4, 8 − 4 x) ,$] find ⓐ #math.equation(block: false, alt: "g open parenthesis −6 close parenthesis")[$g ( −6 )$] ⓑ #math.equation(block: false, alt: "g open parenthesis 2 close parenthesis .")[$g ( 2 ) .$]

#strong[Find the Domain of a Radical Function]

In the following exercises, find the domain of the function and write the domain in interval notation.

#math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the square root of 3 x minus 1")[$f ( x ) = sqrt(3 x − 1)$]

#solutionbox[
#math.equation(block: true, alt: "[ the fraction 1 over 3 , ∞ close parenthesis")[$[ frac(1, 3) , ∞ )$]
]

#math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the square root of 4 x minus 2")[$f ( x ) = sqrt(4 x − 2)$]

#math.equation(block: true, alt: "g open parenthesis x close parenthesis equals the square root of 2 minus 3 x")[$g ( x ) = sqrt(2 − 3 x)$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis − ∞ , the fraction 2 over 3 ]")[$( "−" ∞ , frac(2, 3) ]$]
]

#math.equation(block: true, alt: "g open parenthesis x close parenthesis equals the square root of 8 minus x")[$g ( x ) = sqrt(8 − x)$]

#math.equation(block: true, alt: "h open parenthesis x close parenthesis equals the square root of the fraction 5 over x minus 2")[$h ( x ) = sqrt(frac(5, x − 2))$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 2 , ∞ close parenthesis")[$( 2 , ∞ )$]
]

#math.equation(block: true, alt: "h open parenthesis x close parenthesis equals the square root of the fraction 6 over x plus 3")[$h ( x ) = sqrt(frac(6, x + 3))$]

#math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the square root of the fraction x plus 3 over x minus 2")[$f ( x ) = sqrt(frac(x + 3, x − 2))$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis − ∞ , −3 ] union open parenthesis 2 , ∞ close parenthesis")[$( "−" ∞ , −3 ] ∪ ( 2 , ∞ )$]
]

#math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the square root of the fraction x minus 1 over x plus 4")[$f ( x ) = sqrt(frac(x − 1, x + 4))$]

#math.equation(block: true, alt: "g open parenthesis x close parenthesis equals the cube root of 8 x minus 1")[$g ( x ) = root(3, 8 x − 1)$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis − ∞ , ∞ close parenthesis")[$( "−" ∞ , ∞ )$]
]

#math.equation(block: true, alt: "g open parenthesis x close parenthesis equals the cube root of 6 x plus 5")[$g ( x ) = root(3, 6 x + 5)$]

#math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the cube root of 4 x squared minus 16")[$f ( x ) = root(3, 4 x^(2) − 16)$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis − ∞ , ∞ close parenthesis")[$( "−" ∞ , ∞ )$]
]

#math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the cube root of 6 x squared minus 25")[$f ( x ) = root(3, 6 x^(2) − 25)$]

#math.equation(block: true, alt: "F open parenthesis x close parenthesis equals the 4th root of 8 x plus 3")[$F ( x ) = root(4, 8 x + 3)$]

#solutionbox[
#math.equation(block: true, alt: "[ minus the fraction 3 over 8 , ∞ close parenthesis")[$[ − frac(3, 8) , ∞ )$]
]

#math.equation(block: true, alt: "F open parenthesis x close parenthesis equals the 4th root of 10 minus 7 x")[$F ( x ) = root(4, 10 − 7 x)$]

#math.equation(block: true, alt: "G open parenthesis x close parenthesis equals the 5th root of 2 x minus 1")[$G ( x ) = root(5, 2 x − 1)$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis − ∞ , ∞ close parenthesis")[$( "−" ∞ , ∞ )$]
]

#math.equation(block: true, alt: "G open parenthesis x close parenthesis equals the 5th root of 6 x minus 3")[$G ( x ) = root(5, 6 x − 3)$]

#strong[Graph Radical Functions]

In the following exercises, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

#math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the square root of x plus 1")[$f ( x ) = sqrt(x + 1)$]

#solutionbox[
ⓐ domain: #math.equation(block: false, alt: "[ −1 , ∞ close parenthesis")[$[ −1 , ∞ )$] #linebreak() 
ⓑ  #linebreak() 
#figure(figph[The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 1 to 7. The y-axis runs from negative 2 to 10. The function has a starting point at (negative 1, 0) and goes through the points (0, 1) and (3, 2).], alt: "The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 1 to 7. The y-axis runs from negative 2 to 10. The function has a starting point at (negative 1, 0) and goes through the points (0, 1) and (3, 2).", caption: none)

 #linebreak() 
ⓒ #math.equation(block: false, alt: "[ 0 , ∞ close parenthesis")[$[ 0 , ∞ )$]
]

#math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the square root of x minus 1")[$f ( x ) = sqrt(x − 1)$]

#math.equation(block: true, alt: "g open parenthesis x close parenthesis equals the square root of x plus 4")[$g ( x ) = sqrt(x + 4)$]

#solutionbox[
ⓐ domain: #math.equation(block: false, alt: "[ −4 , ∞ close parenthesis")[$[ −4 , ∞ )$] #linebreak() 
ⓑ  #linebreak() 
#figure(figph[The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 2 to 6. The function has a starting point at (negative 4, 0) and goes through the points (negative 3, 1) and (0, 2).], alt: "The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 2 to 6. The function has a starting point at (negative 4, 0) and goes through the points (negative 3, 1) and (0, 2).", caption: none)

 #linebreak() 
ⓒ #math.equation(block: false, alt: "[ 0 , ∞ close parenthesis")[$[ 0 , ∞ )$]
]

#math.equation(block: true, alt: "g open parenthesis x close parenthesis equals the square root of x minus 4")[$g ( x ) = sqrt(x − 4)$]

#math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the square root of x plus 2")[$f ( x ) = sqrt(x) + 2$]

#solutionbox[
ⓐ domain: #math.equation(block: false, alt: "[ 0 , ∞ close parenthesis")[$[ 0 , ∞ )$] #linebreak() 
ⓑ  #linebreak() 
#figure(figph[The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from 0 to 8. The y-axis runs from 0 to 8. The function has a starting point at (0, 2) and goes through the points (1, 3) and (4, 4).], alt: "The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from 0 to 8. The y-axis runs from 0 to 8. The function has a starting point at (0, 2) and goes through the points (1, 3) and (4, 4).", caption: none)

 #linebreak() 
ⓒ #math.equation(block: false, alt: "[ 2 , ∞ close parenthesis")[$[ 2 , ∞ )$]
]

#math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the square root of x minus 2")[$f ( x ) = sqrt(x) − 2$]

#math.equation(block: true, alt: "g open parenthesis x close parenthesis equals 2 the square root of x")[$g ( x ) = 2 sqrt(x)$]

#solutionbox[
ⓐ domain: #math.equation(block: false, alt: "[ 0 , ∞ close parenthesis")[$[ 0 , ∞ )$] #linebreak() 
ⓑ  #linebreak() 
#figure(figph[The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from 0 to 8. The y-axis runs from 0 to 8. The function has a starting point at (0, 0) and goes through the points (1, 2) and (4, 4).], alt: "The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from 0 to 8. The y-axis runs from 0 to 8. The function has a starting point at (0, 0) and goes through the points (1, 2) and (4, 4).", caption: none)

 #linebreak() 
ⓒ #math.equation(block: false, alt: "[ 0 , ∞ close parenthesis")[$[ 0 , ∞ )$]
]

#math.equation(block: true, alt: "g open parenthesis x close parenthesis equals 3 the square root of x")[$g ( x ) = 3 sqrt(x)$]

#math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the square root of 3 minus x")[$f ( x ) = sqrt(3 − x)$]

#solutionbox[
ⓐ domain: #math.equation(block: false, alt: "open parenthesis − ∞ , 3 ]")[$( "−" ∞ , 3 ]$] #linebreak() 
ⓑ  #linebreak() 
#figure(figph[The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 6 to 4. The y-axis runs from 0 to 8. The function has a starting point at (3, 0) and goes through the points (2, 1), (negative 1, 2), and (negative 6, 3).], alt: "The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 6 to 4. The y-axis runs from 0 to 8. The function has a starting point at (3, 0) and goes through the points (2, 1), (negative 1, 2), and (negative 6, 3).", caption: none)

 #linebreak() 
ⓒ #math.equation(block: false, alt: "[ 0 , ∞ close parenthesis")[$[ 0 , ∞ )$]
]

#math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the square root of 4 minus x")[$f ( x ) = sqrt(4 − x)$]

#math.equation(block: true, alt: "g open parenthesis x close parenthesis equals − the square root of x")[$g ( x ) = "−" sqrt(x)$]

#solutionbox[
ⓐ domain: #math.equation(block: false, alt: "[ 0 , ∞ close parenthesis")[$[ 0 , ∞ )$] #linebreak() 
ⓑ  #linebreak() 
#figure(figph[The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from 0 to 8. The y-axis runs from negative 8 to 0. The function has a starting point at (0, 0) and goes through the points (1, negative 1) and (4, negative 2).], alt: "The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from 0 to 8. The y-axis runs from negative 8 to 0. The function has a starting point at (0, 0) and goes through the points (1, negative 1) and (4, negative 2).", caption: none)

 #linebreak() 
ⓒ #math.equation(block: false, alt: "open parenthesis − ∞ , 0 ]")[$( "−" ∞ , 0 ]$]
]

#math.equation(block: true, alt: "g open parenthesis x close parenthesis equals − the square root of x plus 1")[$g ( x ) = "−" sqrt(x) + 1$]

#math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the cube root of x plus 1")[$f ( x ) = root(3, x + 1)$]

#solutionbox[
ⓐ domain: #math.equation(block: false, alt: "open parenthesis − ∞ , ∞ close parenthesis")[$( "−" ∞ , ∞ )$] #linebreak() 
ⓑ  #linebreak() 
#figure(figph[The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 4 to 4. The function has a center point at (negative 1, 0) and goes through the points (negative 2, negative 1) and (0, 1).], alt: "The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 4 to 4. The function has a center point at (negative 1, 0) and goes through the points (negative 2, negative 1) and (0, 1).", caption: none)

 #linebreak() 
ⓒ #math.equation(block: false, alt: "open parenthesis − ∞ , ∞ close parenthesis")[$( "−" ∞ , ∞ )$]
]

#math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the cube root of x minus 1")[$f ( x ) = root(3, x − 1)$]

#math.equation(block: true, alt: "g open parenthesis x close parenthesis equals the cube root of x plus 2")[$g ( x ) = root(3, x + 2)$]

#solutionbox[
ⓐ domain: #math.equation(block: false, alt: "open parenthesis − ∞ , ∞ close parenthesis")[$( "−" ∞ , ∞ )$] #linebreak() 
ⓑ  #linebreak() 
#figure(figph[The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 4 to 4. The function has a center point at (negative 4, 0) and goes through the points (negative 3, negative 1) and (negative 1, 1).], alt: "The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 4 to 4. The function has a center point at (negative 4, 0) and goes through the points (negative 3, negative 1) and (negative 1, 1).", caption: none)

 #linebreak() 
ⓒ #math.equation(block: false, alt: "open parenthesis − ∞ , ∞ close parenthesis")[$( "−" ∞ , ∞ )$]
]

#math.equation(block: true, alt: "g open parenthesis x close parenthesis equals the cube root of x minus 2")[$g ( x ) = root(3, x − 2)$]

#math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the cube root of x plus 3")[$f ( x ) = root(3, x) + 3$]

#solutionbox[
ⓐ domain: #math.equation(block: false, alt: "open parenthesis − ∞ , ∞ close parenthesis")[$( "−" ∞ , ∞ )$] #linebreak() 
ⓑ  #linebreak() 
#figure(figph[The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 2 to 6. The function has a center point at (0, 3) and goes through the points (negative 1, 2) and (1, 4).], alt: "The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 2 to 6. The function has a center point at (0, 3) and goes through the points (negative 1, 2) and (1, 4).", caption: none)

 #linebreak() 
ⓒ #math.equation(block: false, alt: "open parenthesis − ∞ , ∞ close parenthesis")[$( "−" ∞ , ∞ )$]
]

#math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the cube root of x minus 3")[$f ( x ) = root(3, x) − 3$]

#math.equation(block: true, alt: "g open parenthesis x close parenthesis equals the cube root of x")[$g ( x ) = root(3, x)$]

#solutionbox[
ⓐ domain: #math.equation(block: false, alt: "open parenthesis − ∞ , ∞ close parenthesis")[$( "−" ∞ , ∞ )$] #linebreak() 
ⓑ  #linebreak() 
#figure(figph[The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 4 to 4. The function has a center point at (0, 0) and goes through the points (1, 1) and (negative 1, negative 1).], alt: "The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 4 to 4. The function has a center point at (0, 0) and goes through the points (1, 1) and (negative 1, negative 1).", caption: none)

 #linebreak() 
ⓒ #math.equation(block: false, alt: "open parenthesis − ∞ , ∞ close parenthesis")[$( "−" ∞ , ∞ )$]
]

#math.equation(block: true, alt: "g open parenthesis x close parenthesis equals − the cube root of x")[$g ( x ) = "−" root(3, x)$]

#math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 2 the cube root of x")[$f ( x ) = 2 root(3, x)$]

#solutionbox[
ⓐ domain: #math.equation(block: false, alt: "open parenthesis − ∞ , ∞ close parenthesis")[$( "−" ∞ , ∞ )$] #linebreak() 
ⓑ  #linebreak() 
#figure(figph[The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 4 to 4. The function has a center point at (0, 0) and goes through the points (1, 2) and (negative 1, negative 2).], alt: "The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 4 to 4. The function has a center point at (0, 0) and goes through the points (1, 2) and (negative 1, negative 2).", caption: none)

 #linebreak() 
ⓒ #math.equation(block: false, alt: "open parenthesis − ∞ , ∞ close parenthesis")[$( "−" ∞ , ∞ )$]
]

#math.equation(block: true, alt: "f open parenthesis x close parenthesis equals −2 the cube root of x")[$f ( x ) = −2 root(3, x)$]


==== Writing Exercises

Explain how to find the domain of a fourth root function.

#solutionbox[
Answers will vary.
]

Explain how to find the domain of a fifth root function.

Explain why #math.equation(block: false, alt: "y equals the cube root of x")[$y = root(3, x)$] is a function.

#solutionbox[
Answers will vary.
]

Explain why the process of finding the domain of a radical function with an even index is different from the process when the index is odd.


==== Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

#figure(figph[The table has 4 columns and 4 rows. The first row is a header row with the headers “I can…”, “Confidently”, “With some help.”, and “No – I don’t get it!”. The first column contains the phrases “evaluate a radical function”, “find the domain of a radical function”, and “graph a radical function”. The other columns are left blank so the learner can indicate their level of understanding.], alt: "The table has 4 columns and 4 rows. The first row is a header row with the headers “I can…”, “Confidently”, “With some help.”, and “No – I don’t get it!”. The first column contains the phrases “evaluate a radical function”, “find the domain of a radical function”, and “graph a radical function”. The other columns are left blank so the learner can indicate their level of understanding.", caption: none)

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?