8.7 Use Radicals in Functions

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 radical function.

The square root function is $f (x) = \sqrt[]{x} .$

The cube root function is $f (x) = \sqrt[3]{x} .$

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

Example 1

For the function $f (x) = \sqrt{2 x - 1},$ find ⓐ $f open parenthesis 5 close parenthesis$ ⓑ $f open parenthesis −2 close parenthesis.$

ⓐ

	$f (x) = \sqrt{2 x - 1}$
To evaluate $f open parenthesis 5 close parenthesis,$ substitute 5 for $x .$	$f (5) = \sqrt{2 \cdot 5 - 1}$
Simplify.	$f (5) = \sqrt{9}$
Take the square root.	$f open parenthesis 5 close parenthesis equals 3$

ⓑ

	$f (x) = \sqrt{2 x - 1}$
To evaluate $f open parenthesis −2 close parenthesis,$ substitute −2 for $x .$	$f (−2) = \sqrt{2 (−2) - 1}$
Simplify.	$f (−2) = \sqrt{−5}$

Since the square root of a negative number is not a real number, the function does not have a value at $x equals −2.$

We follow the same procedure to evaluate cube roots.

Example 2

For the function $g (x) = \sqrt[3]{x - 6},$ find ⓐ $g open parenthesis 14 close parenthesis$ ⓑ $g open parenthesis −2 close parenthesis.$

ⓐ

	$g (x) = \sqrt[3]{x - 6}$
To evaluate $g open parenthesis 14 close parenthesis,$ substitute 14 for $x .$	$g (14) = \sqrt[3]{14 - 6}$
Simplify.	$g (14) = \sqrt[3]{8}$
Take the cube root.	$g open parenthesis 14 close parenthesis equals 2$

ⓑ

	$g (x) = \sqrt[3]{x - 6}$
To evaluate $g open parenthesis −2 close parenthesis,$ substitute −2 for $x .$	$g (−2) = \sqrt[3]{−2 - 6}$
Simplify.	$g (−2) = \sqrt[3]{−8}$
Take the cube root.	$g open parenthesis −2 close parenthesis equals −2$

The next example has fourth roots.

Find the Domain of a Radical Function

To find the domain and 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.

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.

Example 4

Find the domain of the function, $f (x) = \sqrt{3 x - 4} .$ Write the domain in interval notation.

Since the function, $f (x) = \sqrt{3 x - 4}$ 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.

	$3 x minus 4 greater than or equal to 0$
Solve.	$3 x greater than or equal to 4$
	$x \geq \frac{4}{3}$

The domain of $f (x) = \sqrt{3 x - 4}$ is all values $x \geq \frac{4}{3}$ and we write it in interval notation as $[\frac{4}{3}, \infty) .$

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

Graph Radical Functions

Before we graph any radical function, we first find the domain of the function. For the function, $f (x) = \sqrt{x},$ the index is even, and so the radicand must be greater than or equal to 0.

This tells us the domain is $x greater than or equal to 0$ and we write this in interval notation as $[ 0, ∞ close parenthesis.$

Previously we used point plotting to graph the function, $f (x) = \sqrt{x} .$ We chose 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.

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).

Once we see the graph, we can find the range of the function. The y-values of the function are greater than or equal to zero. The range then is $[ 0, ∞ close parenthesis.$

In our previous work graphing functions, we graphed $f (x) = x^{3}$ but we did not graph the function $f (x) = \sqrt[3]{x} .$ We will do this now in the next example.

Key Concepts

Properties of $\sqrt[n]{a}$
- When n is an even number and:
  $a greater than or equal to 0,$ then $\sqrt[n]{a}$ is a real number.
  $a less than 0,$ then $\sqrt[n]{a}$ is not a real number.
- When n is an odd number, $\sqrt[n]{a}$ is a real number for all values of a.
Domain of a Radical Function
- When the index of the radical is even, the radicand must be greater than or equal to zero.
- When the index of the radical is odd, the radicand can be any real number.

Practice Makes Perfect

Evaluate a Radical Function

In the following exercises, evaluate each function.

$f (x) = \sqrt{4 x - 4},$ find ⓐ $f open parenthesis 5 close parenthesis$ ⓑ $f open parenthesis 0 close parenthesis.$

ⓐ $f open parenthesis 5 close parenthesis equals 4$ ⓑ no value at $x equals 0$

$f (x) = \sqrt{6 x - 5},$ find ⓐ $f open parenthesis 5 close parenthesis$ ⓑ $f open parenthesis −1 close parenthesis.$

$g (x) = \sqrt{6 x + 1},$ find ⓐ $g open parenthesis 4 close parenthesis$ ⓑ $g open parenthesis 8 close parenthesis.$

ⓐ $g open parenthesis 4 close parenthesis equals 5$ ⓑ $g open parenthesis 8 close parenthesis equals 7$

$g (x) = \sqrt{3 x + 1},$ find ⓐ $g open parenthesis 8 close parenthesis$ ⓑ $g open parenthesis 5 close parenthesis.$

$F (x) = \sqrt{3 - 2 x},$ find ⓐ $F open parenthesis 1 close parenthesis$ ⓑ $F open parenthesis −11 close parenthesis.$

ⓐ $F open parenthesis 1 close parenthesis equals 1$ ⓑ $F open parenthesis −11 close parenthesis equals 5$

$F (x) = \sqrt{8 - 4 x},$ find ⓐ $F open parenthesis 1 close parenthesis$ ⓑ $F open parenthesis −2 close parenthesis.$

$G (x) = \sqrt{5 x - 1},$ find ⓐ $G open parenthesis 5 close parenthesis$ ⓑ $G open parenthesis 2 close parenthesis.$

ⓐ $G (5) = 2 \sqrt{6}$ ⓑ $G open parenthesis 2 close parenthesis equals 3$

$G (x) = \sqrt{4 x + 1},$ find ⓐ $G open parenthesis 11 close parenthesis$ ⓑ $G open parenthesis 2 close parenthesis.$

$g (x) = \sqrt[3]{2 x - 4},$ find ⓐ $g open parenthesis 6 close parenthesis$ ⓑ $g open parenthesis −2 close parenthesis.$

ⓐ $g open parenthesis 6 close parenthesis equals 2$ ⓑ $g open parenthesis −2 close parenthesis equals −2$

$g (x) = \sqrt[3]{7 x - 1},$ find ⓐ $g open parenthesis 4 close parenthesis$ ⓑ $g open parenthesis −1 close parenthesis.$

$h (x) = \sqrt[3]{x^{2} - 4},$ find ⓐ $h open parenthesis −2 close parenthesis$ ⓑ $h open parenthesis 6 close parenthesis.$

ⓐ $h open parenthesis −2 close parenthesis equals 0$ ⓑ $h (6) = 2 \sqrt[3]{4}$

$h (x) = \sqrt[3]{x^{2} + 4},$ find ⓐ $h open parenthesis −2 close parenthesis$ ⓑ $h open parenthesis 6 close parenthesis.$

For the function $f (x) = \sqrt[4]{2 x^{3}},$ find ⓐ $f open parenthesis 0 close parenthesis$ ⓑ $f open parenthesis 2 close parenthesis.$

ⓐ $f open parenthesis 0 close parenthesis equals 0$ ⓑ $f open parenthesis 2 close parenthesis equals 2$

For the function $f (x) = \sqrt[4]{3 x^{3}},$ find ⓐ $f open parenthesis 0 close parenthesis$ ⓑ $f open parenthesis 3 close parenthesis.$

For the function $g (x) = \sqrt[4]{4 - 4 x},$ find ⓐ $g open parenthesis 1 close parenthesis$ ⓑ $g open parenthesis −3 close parenthesis.$

ⓐ $g open parenthesis 1 close parenthesis equals 0$ ⓑ $g open parenthesis −3 close parenthesis equals 2$

For the function $g (x) = \sqrt[4]{8 - 4 x},$ find ⓐ $g open parenthesis −6 close parenthesis$ ⓑ $g open parenthesis 2 close parenthesis.$

Find the Domain of a Radical Function

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

$f (x) = \sqrt{3 x - 1}$

$[\frac{1}{3}, \infty)$

$f (x) = \sqrt{4 x - 2}$

$g (x) = \sqrt{2 - 3 x}$

$(− \infty, \frac{2}{3}]$

$g (x) = \sqrt{8 - x}$

$h (x) = \sqrt{\frac{5}{x - 2}}$

$open parenthesis 2, ∞ close parenthesis$

$h (x) = \sqrt{\frac{6}{x + 3}}$

$f (x) = \sqrt{\frac{x + 3}{x - 2}}$

$open parenthesis − ∞, −3 ] union open parenthesis 2, ∞ close parenthesis$

$f (x) = \sqrt{\frac{x - 1}{x + 4}}$

$g (x) = \sqrt[3]{8 x - 1}$

$open parenthesis − ∞, ∞ close parenthesis$

$g (x) = \sqrt[3]{6 x + 5}$

$f (x) = \sqrt[3]{4 x^{2} - 16}$

$open parenthesis − ∞, ∞ close parenthesis$

$f (x) = \sqrt[3]{6 x^{2} - 25}$

$F (x) = \sqrt[4]{8 x + 3}$

$[- \frac{3}{8}, \infty)$

$F (x) = \sqrt[4]{10 - 7 x}$

$G (x) = \sqrt[5]{2 x - 1}$

$open parenthesis − ∞, ∞ close parenthesis$

$G (x) = \sqrt[5]{6 x - 3}$

Graph Radical Functions

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

$f (x) = \sqrt{x + 1}$

ⓐ domain: $[ −1, ∞ close parenthesis$
ⓑ

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).

ⓒ

[ 0, ∞ close parenthesis

$f (x) = \sqrt{x - 1}$

$g (x) = \sqrt{x + 4}$

ⓐ domain: $[ −4, ∞ close parenthesis$
ⓑ

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).

ⓒ

[ 0, ∞ close parenthesis

$g (x) = \sqrt{x - 4}$

$f (x) = \sqrt{x} + 2$

ⓐ domain: $[ 0, ∞ close parenthesis$
ⓑ

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).

ⓒ

[ 2, ∞ close parenthesis

$f (x) = \sqrt{x} - 2$

$g (x) = 2 \sqrt{x}$

ⓐ domain: $[ 0, ∞ close parenthesis$
ⓑ

ⓒ

[ 0, ∞ close parenthesis

$g (x) = 3 \sqrt{x}$

$f (x) = \sqrt{3 - x}$

ⓐ domain: $open parenthesis − ∞, 3 ]$
ⓑ

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).

ⓒ

[ 0, ∞ close parenthesis

$f (x) = \sqrt{4 - x}$

$g (x) = − \sqrt{x}$

ⓐ domain: $[ 0, ∞ close parenthesis$
ⓑ

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).

ⓒ

open parenthesis − ∞, 0 ]

$g (x) = − \sqrt{x} + 1$

$f (x) = \sqrt[3]{x + 1}$