5.1 Function Composition

Exercises

In Exercises -, use the given pair of functions to find the following values if they exist.

• $(g\circ f)(0)$
• $(f\circ g)(-1)$
• $(f\circ f)(2)$

• $(g\circ f)(-3)$
• $(f\circ g)\left(\frac{1}{2}\right)$
• $(f\circ f)(-2)$

1. $f(x)=x^2$, $g(x)=2x+1$
2. $f(x)=4-x$, $g(x)=1-x^2$
3. $f(x)=4-3x$, $g(x)=|x|$
4. $f(x)=|x-1|$, $g(x)=x^2-5$
5. $f(x)=4x+5$, $g(x)=\sqrt{x}$
6. $f(x)=\sqrt{3-x}$, $g(x)=x^2+1$
7. $f(x)=6-x-x^2$, $g(x)=x\sqrt{x+10}$
8. $f(x)=\sqrt[3]{x+1}$, $g(x)=4x^2-x$
9. $f(x)=\frac{3}{1-x}$, $g(x)=\frac{4x}{x^2+1}$
10. $f(x)=\frac{x}{x+5}$, $g(x)=\frac{2}{7-x^2}$
11. $f(x)=\frac{2x}{5-x^2}$, $g(x)=\sqrt{4x+1}$
12. $f(x)=\sqrt{2x+5}$, $g(x)=\frac{10x}{x^2+1}$

In Exercises -, use the given pair of functions to find and simplify expressions for the following functions and state the domain of each using interval notation.

• $(g\circ f)(x)$
• $(f\circ g)(x)$
• $(f\circ f)(x)$
• $f(x)=2x+3$, $g(x)=x^2-9$
• $f(x)=x^2-x+1$, $g(x)=3x-5$
• $f(x)=x^2-4$, $g(x)=|x|$
• $f(x)=3x-5$, $g(x)=\sqrt{x}$
• $f(x)=|x+1|$, $g(x)=\sqrt{x}$
• $f(x)=3-x^2$, $g(x)=\sqrt{x+1}$
• $f(x)=|x|$, $g(x)=\sqrt{4-x}$
• $f(x)=x^2-x-1$, $g(x)=\sqrt{x-5}$
• $f(x)=3x-1$, $g(x)=\frac{1}{x+3}$
• $f(x)=\frac{3x}{x-1}$, $g(x)=\frac{x}{x-3}$
• $f(x)=\frac{x}{2x+1}$, $g(x)=\frac{2x+1}{x}$
• $f(x)=\frac{2x}{x^2-4}$, $g(x)=\sqrt{1-x}$
• $(h\circ g\circ f)(x)$
• $(h\circ f\circ g)(x)$
• $(g\circ f\circ h)(x)$
• $(g\circ h\circ f)(x)$
• $(f\circ h\circ g)(x)$
• $(f\circ g\circ h)(x)$
• $p(x)={(2x+3)}^3$
• $P(x)={\left(x^2-x+1\right)}^5$
• $h(x)=\sqrt{2x-1}$
• $H(x)=|7-3x|$
• $r(x)=\frac{2}{5x+1}$
• $R(x)=\frac{7}{x^2-1}$
• $q(x)=\frac{|x|+1}{|x|-1}$
• $Q(x)=\frac{2x^3+1}{x^3-1}$
• $v(x)=\frac{2x+1}{3-4x}$
• $w(x)=\frac{x^2}{x^4+1}$
• Write the function $F(x)=\sqrt{\frac{x^3+6}{x^3-9}}$ as a composition of three or more non-identity functions.
• Let $$g(x)=-x,h(x)=x+2,j(x)=3x$$ and $k(x)=x-4$. In what order must these functions be composed with $f(x)=\sqrt{x}$ to create $F(x)=3\sqrt{-x+2}-4$?
• What linear functions could be used to transform $f(x)=x^3$ into $F(x)=-\frac{1}{2}{(2x-7)}^3+1$? What is the proper order of composition?

In Exercises -, use $f(x)=-2x$, $g(x)=\sqrt{x}$ and $h(x)=|x|$ to find and simplify expressions for the following functions and state the domain of each using interval notation.

In Exercises -, write the given function as a composition of two or more non-identity functions. (There are several correct answers, so check your answer using function composition.)

In Exercises -, let $f$ be the function defined by

$$f=\{(-3,4),(-2,2),(-1,0),(0,1),(1,3),(2,4),(3,-1)\}$$

and let $g$ be the function defined

$$g=\{(-3,-2),(-2,0),(-1,-4),(0,0),(1,-3),(2,1),(3,2)\}$$

. Find the value if it exists.

1. $(f\circ g)(3)$
2. $f(g(-1))$
3. $(f\circ f)(0)$
4. $(f\circ g)(-3)$
5. $(g\circ f)(3)$
6. $g(f(-3))$
7. $(g\circ g)(-2)$
8. $(g\circ f)(-2)$
9. $g(f(g(0)))$
10. $f(f(f(-1)))$
11. $f(f(f(f(f(1)))))$
12. $\underset{n\text{ times}}{\underbrace{(g\circ g\circ \mathrm{\cdots }\circ g)}}(0)$

In Exercises -, use the graphs of $y=f(x)$ and $y=g(x)$ below to find the function value.

1. $(g\circ f)(1)$
2. $(f\circ g)(3)$
3. $(g\circ f)(2)$
4. $(f\circ g)(0)$
5. $(f\circ f)(1)$
6. $(g\circ g)(1)$
7. The volume $V$ of a cube is a function of its side length $x$. Let's assume that $x=t+1$ is also a function of time $t$, where $x$ is measured in inches and $t$ is measured in minutes. Find a formula for $V$ as a function of $t$.

8. Suppose a local vendor charges $\$2$ per hot dog and that the number of hot dogs sold per hour $x$ is given by $x(t)=-4t^2+20t+92$, where $t$ is the number of hours since $10$ AM, $0\le t\le 4$.

   1. Find an expression for the revenue per hour $R$ as a function of $x$.
   2. Find and simplify $\left(R\circ x\right)(t)$. What does this represent?
   3. What is the revenue per hour at noon?
9. Discuss with your classmates how `real-world' processes such as filling out federal income tax forms or computing your final course grade could be viewed as a use of function composition. Find a process for which composition with itself (iteration) makes sense.

Answers

1. For $f(x)=x^2$ and $g(x)=2x+1$,

   • $(g\circ f)(0)=1$
   • $(f\circ g)(-1)=1$
   • $(f\circ f)(2)=16$
   • $(g\circ f)(-3)=19$
   • $(f\circ g)\left(\frac{1}{2}\right)=4$
   • $(f\circ f)(-2)=16$

2. For $f(x)=4-x$ and $g(x)=1-x^2$,

   • $(g\circ f)(0)=-15$
   • $(f\circ g)(-1)=4$
   • $(f\circ f)(2)=2$
   • $(g\circ f)(-3)=-48$
   • $(f\circ g)\left(\frac{1}{2}\right)=\frac{13}{4}$
   • $(f\circ f)(-2)=-2$

3. For $f(x)=4-3x$ and $g(x)=|x|$,

   • $(g\circ f)(0)=4$
   • $(f\circ g)(-1)=1$
   • $(f\circ f)(2)=10$
   • $(g\circ f)(-3)=13$
   • $(f\circ g)\left(\frac{1}{2}\right)=\frac{5}{2}$
   • $(f\circ f)(-2)=-26$

4. For $f(x)=|x-1|$ and $g(x)=x^2-5$,

   • $(g\circ f)(0)=-4$
   • $(f\circ g)(-1)=5$
   • $(f\circ f)(2)=0$
   • $(g\circ f)(-3)=11$
   • $(f\circ g)\left(\frac{1}{2}\right)=\frac{23}{4}$
   • $(f\circ f)(-2)=2$

5. For $f(x)=4x+5$ and $g(x)=\sqrt{x}$,

   • $(g\circ f)(0)=\sqrt{5}$
   • $(f\circ g)(-1)$ is not real
   • $(f\circ f)(2)=57$
   • $(g\circ f)(-3)$ is not real
   • $$(f\circ g)\left(\frac{1}{2}\right)=5+2\sqrt{2}$$
   • $(f\circ f)(-2)=-7$

6. For $f(x)=\sqrt{3-x}$ and $g(x)=x^2+1$,

   • $(g\circ f)(0)=4$
   • $(f\circ g)(-1)=1$
   • $(f\circ f)(2)=\sqrt{2}$
   • $(g\circ f)(-3)=7$
   • $(f\circ g)\left(\frac{1}{2}\right)=\frac{\sqrt{7}}{2}$
   • $$(f\circ f)(-2)=\sqrt{3-\sqrt{5}}$$

7. For $f(x)=6-x-x^2$ and $g(x)=x\sqrt{x+10}$,

   • $(g\circ f)(0)=24$
   • $(f\circ g)(-1)=0$
   • $(f\circ f)(2)=6$
   • $(g\circ f)(-3)=0$
   • $$(f\circ g)\left(\frac{1}{2}\right)=\frac{27-2\sqrt{42}}{8}$$
   • $(f\circ f)(-2)=-14$

8. For $f(x)=\sqrt[3]{x+1}$ and $g(x)=4x^2-x$,

   • $(g\circ f)(0)=3$
   • $(f\circ g)(-1)=\sqrt[3]{6}$
   • $$(f\circ f)(2)=\sqrt[3]{\sqrt[3]{3}+1}$$
   • $$(g\circ f)(-3)=4\sqrt[3]{4}+\sqrt[3]{2}$$
   • $(f\circ g)\left(\frac{1}{2}\right)=\frac{\sqrt[3]{12}}{2}$
   • $(f\circ f)(-2)=0$

9. For $f(x)=\frac{3}{1-x}$ and $g(x)=\frac{4x}{x^2+1}$,

   • $(g\circ f)(0)=\frac{6}{5}$
   • $(f\circ g)(-1)=1$
   • $(f\circ f)(2)=\frac{3}{4}$
   • $(g\circ f)(-3)=\frac{48}{25}$
   • $(f\circ g)\left(\frac{1}{2}\right)=-5$
   • $(f\circ f)(-2)$ is undefined

10. For $f(x)=\frac{x}{x+5}$ and $g(x)=\frac{2}{7-x^2}$,

    • $(g\circ f)(0)=\frac{2}{7}$
    • $(f\circ g)(-1)=\frac{1}{16}$
    • $(f\circ f)(2)=\frac{2}{37}$
    • $(g\circ f)(-3)=\frac{8}{19}$
    • $(f\circ g)\left(\frac{1}{2}\right)=\frac{8}{143}$
    • $(f\circ f)(-2)=-\frac{2}{13}$

11. For $f(x)=\frac{2x}{5-x^2}$ and $g(x)=\sqrt{4x+1}$,

    • $(g\circ f)(0)=1$
    • $(f\circ g)(-1)$ is not real
    • $(f\circ f)(2)=-\frac{8}{11}$
    • $(g\circ f)(-3)=\sqrt{7}$
    • $(f\circ g)\left(\frac{1}{2}\right)=\sqrt{3}$
    • $(f\circ f)(-2)=\frac{8}{11}$

12. For $f(x)=\sqrt{2x+5}$ and $g(x)=\frac{10x}{x^2+1}$,

    • $(g\circ f)(0)=\frac{5\sqrt{5}}{3}$
    • $(f\circ g)(-1)$ is not real
    • $(f\circ f)(2)=\sqrt{11}$
    • $(g\circ f)(-3)$ is not real
    • $(f\circ g)\left(\frac{1}{2}\right)=\sqrt{13}$
    • $(f\circ f)(-2)=\sqrt{7}$

13. For $f(x)=2x+3$ and $g(x)=x^2-9$

    • $$(g\circ f)(x)=4x^2+12x$$, domain: $(-\mathrm{\infty },\mathrm{\infty })$
    • $$(f\circ g)(x)=2x^2-15$$, domain: $(-\mathrm{\infty },\mathrm{\infty })$
    • $(f\circ f)(x)=4x+9$, domain: $(-\mathrm{\infty },\mathrm{\infty })$

14. For $f(x)=x^2-x+1$ and $g(x)=3x-5$

    • $$(g\circ f)(x)=3x^2-3x-2$$, domain: $(-\mathrm{\infty },\mathrm{\infty })$
    • $$(f\circ g)(x)=9x^2-33x+31$$, domain: $(-\mathrm{\infty },\mathrm{\infty })$
    • $$(f\circ f)(x)=x^4-2x^3+2x^2-x+1$$, domain: $(-\mathrm{\infty },\mathrm{\infty })$

15. For $f(x)=x^2-4$ and $g(x)=|x|$

    • $$(g\circ f)(x)=|x^2-4|$$, domain: $(-\mathrm{\infty },\mathrm{\infty })$
    • $$(f\circ g)(x)={|x|}^2-4=x^2-4$$, domain: $(-\mathrm{\infty },\mathrm{\infty })$
    • $$(f\circ f)(x)=x^4-8x^2+12$$, domain: $(-\mathrm{\infty },\mathrm{\infty })$

16. For $f(x)=3x-5$ and $g(x)=\sqrt{x}$

    • $$(g\circ f)(x)=\sqrt{3x-5}$$, domain: $[\frac{5}{3},\mathrm{\infty })$
    • $$(f\circ g)(x)=3\sqrt{x}-5$$, domain: $[0,\mathrm{\infty })$
    • $$(f\circ f)(x)=9x-20$$, domain: $(-\mathrm{\infty },\mathrm{\infty })$

17. For $f(x)=|x+1|$ and $g(x)=\sqrt{x}$

    • $$(g\circ f)(x)=\sqrt{|x+1|}$$, domain: $(-\mathrm{\infty },\mathrm{\infty })$
    • $$(f\circ g)(x)=|\sqrt{x}+1|=\sqrt{x}+1$$, domain: $[0,\mathrm{\infty })$
    • $$(f\circ f)(x)=||x+1|+1|=|x+1|+1$$, domain: $(-\mathrm{\infty },\mathrm{\infty })$

18. For $f(x)=3-x^2$ and $g(x)=\sqrt{x+1}$

    • $$(g\circ f)(x)=\sqrt{4-x^2}$$, domain: $[-2,2]$
    • $(f\circ g)(x)=2-x$, domain: $[-1,\mathrm{\infty })$
    • $$(f\circ f)(x)=-x^4+6x^2-6$$, domain: $(-\mathrm{\infty },\mathrm{\infty })$

19. For $f(x)=|x|$ and $g(x)=\sqrt{4-x}$

    • $$(g\circ f)(x)=\sqrt{4-|x|}$$, domain: $[-4,4]$
    • $$(f\circ g)(x)=|\sqrt{4-x}|=\sqrt{4-x}$$, domain: $(-\mathrm{\infty },4]$
    • $$(f\circ f)(x)=\Vert x\Vert =|x|$$, domain: $(-\mathrm{\infty },\mathrm{\infty })$

20. For $f(x)=x^2-x-1$ and $g(x)=\sqrt{x-5}$

    • $$(g\circ f)(x)=\sqrt{x^2-x-6}$$, domain: $(-\mathrm{\infty },-2]\cup [3,\mathrm{\infty })$
    • $$(f\circ g)(x)=x-6-\sqrt{x-5}$$, domain: $[5,\mathrm{\infty })$
    • $$(f\circ f)(x)=x^4-2x^3-2x^2+3x+1$$, domain: $(-\mathrm{\infty },\mathrm{\infty })$

21. For $f(x)=3x-1$ and $g(x)=\frac{1}{x+3}$

    • $(g\circ f)(x)=\frac{1}{3x+2}$, domain: $(-\mathrm{\infty },-\frac{2}{3})\cup (-\frac{2}{3},\mathrm{\infty })$
    • $(f\circ g)(x)=-\frac{x}{x+3}$, domain: $$(-\mathrm{\infty },-3)\cup (-3,\mathrm{\infty })$$
    • $(f\circ f)(x)=9x-4$, domain: $(-\mathrm{\infty },\mathrm{\infty })$

22. For $f(x)=\frac{3x}{x-1}$ and $g(x)=\frac{x}{x-3}$

    • $(g\circ f)(x)=x$, domain: $(-\mathrm{\infty },1)\cup (1,\mathrm{\infty })$
    • $(f\circ g)(x)=x$, domain: $(-\mathrm{\infty },3)\cup (3,\mathrm{\infty })$
    • $(f\circ f)(x)=\frac{9x}{2x+1}$, domain: $$(-\mathrm{\infty },-\frac{1}{2})\cup (-\frac{1}{2},1)\cup (1,\mathrm{\infty })$$

23. For $f(x)=\frac{x}{2x+1}$ and $g(x)=\frac{2x+1}{x}$

    • $(g\circ f)(x)=\frac{4x+1}{x}$, domain: $$(-\mathrm{\infty },-\frac{1}{2})\cup (-\frac{1}{2},0),\cup (0,\mathrm{\infty })$$
    • $(f\circ g)(x)=\frac{2x+1}{5x+2}$, domain: $$(-\mathrm{\infty },-\frac{2}{5})\cup (-\frac{2}{5},0)\cup (0,\mathrm{\infty })$$
    • $(f\circ f)(x)=\frac{x}{4x+1}$, domain: $$(-\mathrm{\infty },-\frac{1}{2})\cup (-\frac{1}{2},-\frac{1}{4})\cup (-\frac{1}{4},\mathrm{\infty })$$

24. For $f(x)=\frac{2x}{x^2-4}$ and $g(x)=\sqrt{1-x}$

    • $$(g\circ f)(x)=\sqrt{\frac{x^2-2x-4}{x^2-4}}$$, domain: $$(-\mathrm{\infty },-2)\cup [1-\sqrt{5},2)\cup [1+\sqrt{5},\mathrm{\infty })$$
    • $$(f\circ g)(x)=-\frac{2\sqrt{1-x}}{x+3}$$, domain: $(-\mathrm{\infty },-3)\cup (-3,1]$
    • $$(f\circ f)(x)=\frac{4x-x^3}{x^4-9x^2+16}$$, domain: $$(-\mathrm{\infty },-\frac{1+\sqrt{17}}{2})\cup (-\frac{1+\sqrt{17}}{2},-2)\cup (-2,\frac{1-\sqrt{17}}{2})\cup (\frac{1-\sqrt{17}}{2},\frac{-1+\sqrt{17}}{2})\cup (\frac{-1+\sqrt{17}}{2},2)\cup (2,\frac{1+\sqrt{17}}{2})\cup (\frac{1+\sqrt{17}}{2},\mathrm{\infty })$$
25. $$(h\circ g\circ f)(x)=|\sqrt{-2x}|=\sqrt{-2x}$$, domain: $(-\mathrm{\infty },0]$
26. $$(h\circ f\circ g)(x)=|-2\sqrt{x}|=2\sqrt{x}$$, domain: $[0,\mathrm{\infty })$
27. $(g\circ f\circ h)(x)=\sqrt{-2|x|}$, domain: $\{0\}$
28. $$(g\circ h\circ f)(x)=\sqrt{|-2x|}=\sqrt{2|x|}$$, domain: $(-\mathrm{\infty },\mathrm{\infty })$
29. $$(f\circ h\circ g)(x)=-2|\sqrt{x}|=-2\sqrt{x}$$, domain: $[0,\mathrm{\infty })$
30. $(f\circ g\circ h)(x)=-2\sqrt{|x|}$,, domain: $(-\mathrm{\infty },\mathrm{\infty })$
31. Let $f(x)=2x+3$ and $g(x)=x^3$, then $p(x)=(g\circ f)(x)$.
32. Let $f(x)=x^2-x+1$ and $g(x)=x^5$, $P(x)=(g\circ f)(x)$.
33. Let $f(x)=2x-1$ and $g(x)=\sqrt{x}$, then $h(x)=(g\circ f)(x)$.
34. Let $f(x)=7-3x$ and $g(x)=|x|$, then $H(x)=(g\circ f)(x)$.
35. Let $f(x)=5x+1$ and $g(x)=\frac{2}{x}$, then $r(x)=(g\circ f)(x)$.
36. Let $f(x)=x^2-1$ and $g(x)=\frac{7}{x}$, then $R(x)=(g\circ f)(x)$.
37. Let $f(x)=|x|$ and $g(x)=\frac{x+1}{x-1}$, then $q(x)=(g\circ f)(x)$.
38. Let $f(x)=x^3$ and $g(x)=\frac{2x+1}{x-1}$, then $Q(x)=(g\circ f)(x)$.
39. Let $f(x)=2x$ and $g(x)=\frac{x+1}{3-2x}$, then $v(x)=(g\circ f)(x)$.
40. Let $f(x)=x^2$ and $g(x)=\frac{x}{x^2+1}$, then $w(x)=(g\circ f)(x)$.
41. $$F(x)=\sqrt{\frac{x^3+6}{x^3-9}}=(h(g(f(x)))$$ where $f(x)=x^3,g(x)=\frac{x+6}{x-9}$ and $h(x)=\sqrt{x}$.
42. $$F(x)=3\sqrt{-x+2}-4=k(j(f(h(g(x)))))$$
43. One possible solution is $$F(x)=-\frac{1}{2}{(2x-7)}^3+1=k(j(f(h(g(x)))))$$ where $$g(x)=2x,h(x)=x-7,j(x)=-\frac{1}{2}x$$ and $k(x)=x+1$. You could also have $F(x)=H(f(G(x)))$ where $G(x)=2x-7$ and $H(x)=-\frac{1}{2}x+1$.
44. $$(f\circ g)(3)=f(g(3))=f(2)=4$$
45. $f(g(-1))=f(-4)$ which is undefined
46. $$(f\circ f)(0)=f(f(0))=f(1)=3$$
47. $$(f\circ g)(-3)=f(g(-3))=f(-2)=2$$
48. $$(g\circ f)(3)=g(f(3))=g(-1)=-4$$
49. $g(f(-3))=g(4)$ which is undefined
50. $$(g\circ g)(-2)=g(g(-2))=g(0)=0$$
51. $$(g\circ f)(-2)=g(f(-2))=g(2)=1$$
52. $$g(f(g(0)))=g(f(0))=g(1)=-3$$
53. $$f(f(f(-1)))=f(f(0))=f(1)=3$$
54. $$f(f(f(f(f(1)))))=f(f(f(f(3))))=f(f(f(-1)))=f(f(0))=f(1)=3$$
55. $\underset{n\text{ times}}{\underbrace{(g\circ g\circ \mathrm{\cdots }\circ g)}}(0)=0$
56. $(g\circ f)(1)=3$
57. $(f\circ g)(3)=4$
58. $(g\circ f)(2)=0$
59. $(f\circ g)(0)=4$
60. $(f\circ f)(1)=3$
61. $(g\circ g)(1)=0$
62. $V(x)=x^3$ so $V(x(t))={(t+1)}^3$
63. 1. $R(x)=2x$
    2. $$\left(R\circ x\right)(t)=-8t^2+40t+184$$, $0\le t\le 4$. This gives the revenue per hour as a function of time.
    3. Noon corresponds to $t=2$, so $\left(R\circ x\right)(2)=232$. The hourly revenue at noon is $\$232$ per hour.