Deck 10: Compositions, Inverses, and Combinations of Functions

Question 1

If $h(x)=f(g(x))$ , what is the value of $F$ in the following tables?

Answer 1

9

Question 2

Let $f(x)=2^{x}$ and $g(x)=2 f(f(x))$ . Evaluate $g(0)$ .

Answer 2

4

Question 3

The functions $m(x)$ and $n(x)$ are defined by the graph below. The dashed graph is $m(x)$ and the solid graph is $n(x)$ .

Evaluate $m(n(-3))$ .

Answer 3

-1

Question 4

If $H(x)=f(g(x))=e^{4 x+2}$ , which of the following could be true? Mark all that apply.

A) $f(x)=e^{x}, g(x)=4 x+2$
B) $f(x)=4 x+2, g(x)=e^{x}$
C) $f(x)=x^{2}, g(x)=e^{2 x+1}$
D) $f(x)=e^{2 x+1}, g(x)=x^{2}$
E) $f(x)=x+2, g(x)=e^{4 x}$
F) $f(x)=e^{4 x}, g(x)=x+2$

Question 5

The functions $m(x)$ and $n(x)$ are defined by the graph below. The dashed graph is $m(x)$ and the solid graph is $n(x)$ .

If $p(x)=-2 m(x+3)+6$ , what is $p(-2)$ ?

Question 6

Given the graphs of $f$ and $g$ below, evaluate $f(g(-3))$ .

Question 7

Let $m(x)=f(g(x))$ and $n(x)=g(f(x))$ . Using the table below gives $m(7)=$ -----------and $n(7)=$ -----------------

Question 8

Given $f(x)=\frac{5}{x}$ and $g(x)=x^{2}-1$ , what is $g(f(x)$ ?

A) $\frac{25}{x^{2}}-1$
B) $\frac{5}{x^{2}-1}$
C) $\frac{5}{x^{2}}-1$
D) $\frac{25}{x^{2}-1}$

Question 9

Let $m(x)=e^{x}$ and $n(x)=\frac{x^{4}}{x+5}$ . Does $m(n(x))=\frac{e^{4 x}}{e^{x}+5}$ ?

Question 10

If $u(v(x))=\frac{4}{1+\sqrt{x}}$ , which of the following could be true? Mark all that apply.

A) $u(x)=1+\sqrt{x}, v(x)=\frac{4}{x}$
B) $u(x)=\frac{4}{x}, v(x)=1+\sqrt{x}$
C) $u(x)=\frac{4}{1+x}, v(x)=\sqrt{x}$
D) $u(x)=\sqrt{x}, v(x)=\frac{4}{1+x}$

Question 11

If $f(x)=x^{2}+2 x$ and $g(x)=2-x$ , does $f(g(x))=x^{2}-2 x+4$ ?

Question 12

If $p(q(x))=\frac{4}{1+x}$ and $q(x)=5+x$ , what is $p(x)$ ?

A) $\frac{4}{1+x}$
B) $\frac{4}{1+x}-5$
C) $\frac{4}{-4+x}$
D) $\frac{4}{x}+-4$

Question 13

Let $f(x)=x+8$ and $h(x)=\sqrt{x-5}$ . What is $(h(f(x)))^{2}$ ?

A) $x+3$
B) $x-5$
C) $x+59$
D) $x+59+16 \sqrt{x-5}$

Question 14

Let $f(x)=\cos (7 x)$ and $g(x)=\sqrt{1-x^{2}}$ . What is $g(f(x))$ ?

A) $\sin (7 x)$
B) $-\sin (\sqrt{7} x)$
C) $\sqrt{\cos (49 x)-1}$
D) $\cos \left(7 \sqrt{x^{2}-1}\right)$

Question 15

Let $f$ be defined by the following graph.

Describe the graph of $y=(-2) f(x)$ .

A) A version of the graph of $f$ reflected across the $x$ -axis, stretched vertically by a factor of 2.
B) A version of the graph of $f$ reflected across the $x$ -axis, compressed by a factor of $1 /2$ .
C) A version of the graph of $f$ reflected across the $y$ -axis, stretched vertically by a factor of 2.
D) A version of the graph of $f$ reflected across the $y$ -axis, compressed by a factor of 2..

Question 16

Let $f$ be defined by the following graph.

Describe the graph of $y=f(-x)-2$ .

A) A version of the graph of $f$ flipped about the $y$ -axis and shifted up by 2 units.
B) A version of the graph of $f$ flipped about the $y$ -axis and shifted down by 2 units.
C) A version of the graph of $f$ flipped about the $x$ -axis and shifted down by 2 units.
D) A version of the graph of $f$ flipped about the $x$ -axis and shifted up by 2 units.

Question 17

Let $f(x)=x^{2}+5$ . Does $f(x+h)=x^{2}+h^{2}+5$ ?

Question 18

Let $f(x)=x^{2}+3$ , and $g(x)=\frac{1}{x}+1$ . Does $g(f(x))=\frac{1 x^{2}+1}{x^{2}+3} ?$

Question 19

Let $f(x)=x^{2}+1$ and $h(x)=\sqrt{x-3}$ . Does $h(f(x))=\sqrt{x^{2}-3}$ ?

Question 20

Let $f(x)=x^{2}+1$ and $h(x)=\sqrt{x-5}$ . Does $f(h(x))=x-24$ ?

Question 21

Find $f(f(0))$ for $f(x)=\left\{\begin{array}{cc}0 & x \leq 3 \\ x & 3<x<8 \\ 8 & x \geq 8\end{array}\right.$

Question 22

If $y=\frac{3 x^{2}+1}{6 x^{2}}=u(v(x))$ , which of the following could be true? Mark all that apply.

A) $u(x)=\frac{3 x+1}{6 x}, v(x)=x^{2}$
B) $u(x)=x^{2}, v(x)=\frac{3 x+1}{6 x}$
C) $u(x)=\frac{x}{2 x-2}, v(x)=3 x^{2}+1$
D) $u(x)=\frac{x}{6 x-1}, v(x)=3 x^{2}+1$
E) $u(x)=\frac{3 x+1}{x}, v(x)=6 x^{2}$

Question 23

A child building a tower with blocks places 18 blocks in the first row, 17 blocks in the second row, 16 blocks in the third row, and so forth. How many blocks are in the $10^{\text {th }}$ row?

Question 24

Select the list of functions that is a decomposition of $f(x)=\csc ^{3}(\ln x)$ in the form of $g(h(p(x)))$ .

A) $g(x)=\ln x, h(x)=\csc x, p(x)=x^{3}$
B) $g(x)=\csc x, h(x)=x^{3}, p(x)=\ln x$
C) $g(x)=x^{3}, h(x)=\csc x, p(x)=\ln x$
D) $g(x)=3^{x}, h(x)=\csc x, p(x)=\ln x$

Question 25

Let $f(x)=\frac{1}{7+x}$ . Does $f^{-1}(x)=\frac{1+7 x}{x}$ ?

Question 26

Let $f(x)=\frac{4}{2+x}$ and let $g(x)=f(f(x))$ . What is $g^{-1}(x)$ ?

A) $\frac{8-4 x}{8-4 x}$
B) $\frac{8+4 x}{8+4 x}$
C) $\frac{8+8 x}{-4-2 x}$
D) $\frac{8-8 x}{-4+2 x}$

Question 27

Given $f^{-1}(x)=400(1.02)^{x}$ , what is $f(x)$ ?

A) $\frac{\sqrt[x]{1.02}}{400}$
B) $1.02 \sqrt[x]{400}$
C) $\frac{\ln (x)-\ln (1.02)}{\ln (400)}$
D) $\frac{\ln (x)-\ln (400)}{\ln (1.02)}$

Question 28

Let $f(x)=2^{x}$ and $g(x)=f(f(x))$ . Evaluate $g^{-1}(16)$ .

Question 29

Let $f(x)=2^{x}$ and $g(x)=f(f(x))$ . Evaluate $g^{-1}(11)$ to 2 decimal places.

Question 30

Given the graph of $g$ below, which of the following is a solution to the equation $g(x)=1$ ?

A) 1
B) 0
C) -3
D) 3
E) 2

Question 31

Given the graph of $f$ below, solve $f(x)=x$ . If there is more than one answer, enter them from least to greatest, separated by semicolons. If there are no solutions, enter "none".

Question 32

Based on the following table, could the function $f(x)$ be invertible?

Question 33

Is the function $y=\left|\frac{4}{x}\right|$ invertible if the domain of $f$ is all real numbers?

Question 34

Is the function $f(x)=\frac{x^{3}+4}{4}$ the inverse of the function $g(t)=\sqrt[3]{4 t-4}$ ?

Question 35

For positive numbers $x$ , what is the inverse of $h(x)=e^{\sqrt{x}-2}$ ?

A) $(\ln (x+2))^{2}$
B) $(\ln x)^{2}+2$
C) $\ln \left((x+2)^{2}\right)$
D) $(\ln x+2)^{2}$

Question 36

What is the inverse of $f(x)=\frac{4 x+1}{4 x-1}$ ?

A) $\frac{1 x+1}{4 x-4}$
B) $\frac{1 x-1}{4 x+4}$
C) $\frac{4 x-1}{4 x+1}$
D) $\frac{-4 x-1}{-4 x+1}$

Question 37

Given $f^{-1}(x)=800(1.05)^{x}$ , solve $f^{-1}(x)=1000$ . Round to 3 decimal places.

Question 38

Given $f^{-1}(x)=1,300(1.03)^{x}$ , solve $f(x)=0$ .

Question 39

Given $f(x)=\frac{5+\ln (x-3)}{5-\ln (x-3)}$ , does $f^{-1}(x)=e^{\frac{5 x-5}{x+1}}+3$ ?

Question 40

Let $g(x)=\frac{6}{x}+6$ . Does $g^{-1}(x)=\frac{6}{x-6}$ ?

Question 41

Is the function graphed below invertible?

Question 42

Use a graph to determine whether or not $y=6 x^{5}+3$ is invertible.

Question 43

Let $P=20 \ln (t)$ give the annual profit of a company (in thousands of dollars) $t$ years after its formation. What is $P^{-1}(38)$ ? Round to the nearest whole number and include units.

Question 44

The following figure defines a function $f$ .

Which of the following quantities is greater?

A) $f(2)$
B) $f^{-1}(3)$

Question 45

Suppose $f(x)=\sqrt{8-x}$ . What is the domain of $f^{-1}(x)$ ?

A) All real numbers greater than or equal to 8 .
B) All real numbers less than or equal to 8 .
C) All real numbers greater than or equal to 0 .
D) All real numbers less than or equal to 0 .

Question 46

Let $f(x)$ be a linear function with a positive slope and $g(x)$ be a different linear function with a negative slope. Describe the graph of $h(x)=f(x) * g(x)$ .

A) A pair of curves that approach both a horizontal and a vertical asymptote.
B) A parabola that opens upward.
C) Another linear function.
D) A parabola that opens downward.

Question 47

Let $f(x)=\sqrt{a x+1}$ and $g(x)=\ln (b x)$ for positive constants $a$ and $b$ . Let $h(x)=a x \ln \left(b^{2} x^{2}\right)+\ln \left(b^{2} x^{2}\right)$ . If $f(4)=2$ and $g(4)=3$ , what is $h(4) ?$

Question 48

Is the quotient of two even functions; odd, even, or neither?

Question 49

For $f(x)=\frac{6}{x}$ , does $\frac{f(x+h)-f(x)}{h}=\frac{6}{x^{2}+h x}$ ?

Question 50

The following table gives 3 functions, $f, g$ , and $h$ .

Which of the following statements is true?

A) $f(x)=4 x+6$
B) $g(x)=(f(x))^{2}$
C) $g(x)=48 x+100$
D) $h(x)=f(x)+g(x)$
E) $h(x)=(f(x))^{2}-2$

Question 51

The function $h(x)$ is shown in the first figure.

Which transformation of $h(x)$ is shown in the second figure?

A) $y=1 /h(x)$
B) $y=-h(x)$
C) $y=|h(x)|$
D) $y=h(-x)$

Question 52

The following table gives values for the functions $f, g$ , and $h$ , three functions defined only for the values $x=0,1, \ldots, 5$ . Based on the table, what is $2 f(3)+g(h(3))$ ?

Question 53

Let $f(x)=x^{2}+5$ and $g(x)=\frac{1}{x}+2$ . What is $f(4)+g(1)$ ?

Question 54

The functions $m(x)$ and $n(x)$ are defined by the graph below. The dashed graph is $m(x)$ and the solid graph is $n(x)$ .

Evaluate $m(-3) \cdot n(-3)$ .

Question 55

Let $f(x)=x+3$ and $g(x)=x^{2}$ . What is $4 f(x)-g(x)$ ?

A) $-x^{2}+4 x+4$
B) $-x^{2}+4 x+12$
C) $-4 x^{2}+4 x+12$
D) $-4 x^{2}+4 x+4$

Question 56

Let $f(x)=x+1$ and $g(x)=x^{2}$ . What is $f(x) g(x)$ ?

A) $1 x^{2}+x$
B) $x^{3}+1$
C) $x^{3}+1 x^{2}$
D) $x^{2}+x+1$

Question 57

Let $f(x)=x+5, g(x)=x^{5}$ , and $h(x)=\sqrt{x-4}$ . What is $\frac{f(x)}{g(h(x))}$ ?

A) $\frac{x+5}{\sqrt{x^{5}-1024}}$
B) $\frac{x+5}{\sqrt{x^{5}-4}}$
C) $\frac{(x+5)^{5}}{(x-4)^{5 / 2}}$
D) $\frac{x+5}{(x-4)^{5 / 2}}$

Question 58

Find $f(x)(g(x))^{2}$ if $f(x)=5 e^{2 x}+1$ and $g(x)=\frac{1}{7} e^{-x}$ .

Question 59

Let $f(x)=\left(x^{2}+1\right)(x+2)$ and $g(x)=(x-1)\left(x^{2}-4\right)$ . Find a simplified formula for $h(x)=\frac{5 f(x)}{17 g(x)}$ .

Question 60

Let $v(x)=4 e^{x}+3 x$ and $u(x)=-2 x$ . Find a simplified formula for the function $w(x)=(u(v(x)))^{2}$ .

Question 61

Let $f(t)$ be the number of men and $g(t)$ be the number of women residing in a certain town in year $t$ . Let $h(t)$ be the average income, in dollars, of residents of that town in year $t$ . If $f(t)=250+8 t, g(t)=275+5 t$ , and $h(t)=32,000+100 t$ , find a simplified formula for the total amount of money earned by all adult residents of the town in year $t$ .

Question 62

Let $f(t)$ be the number of men and let $g(t)$ be the number of women residing in a certain town in year $t$ . Let $h(t)$ be the average income, in dollars, of residents of that town in year $t$ . If $f(t)=250+5 t, g(t)=275+7 t$ , and $h(t)=32,000+200 t$ , find the total amount of money earned by all adult residents of the town in year 3 .

Question 63

Find a simplified formula for $h$ given
$F(x)=4 x^{3} f(x)=12 x^{2}$
$G(x)=e^{5 x} g(x)=5 e^{5 x}$
$h(x)=\frac{f(x) G(x)-F(x) g(x)}{(G(x))^{2}}$

Question 64

Let $f(x)=\frac{4}{x+1}$ . Find and simplify $f\left(\frac{1}{x}\right)-\frac{1}{f(x)}$ .

Question 65

Find the function $p(x)=f(x)+g(x)$ when $f(x)=4 x+1$ and $g(x)=2 x+6$ .

Question 66

Find the function $p(x)=f(x)-g(x)$ when $f(x)=6 x+1$ and $g(x)=3 x+6$ .

Question 67

Find the function $p(x)=f(x)+g(x)$ when $f(x)=4 x^{3}+5$ and $g(x)=3 x+2$ .

Question 68

Find the function $p(x)=f(x)-g(x)$ when $f(x)=4 x^{3}+7$ and $g(x)=2 x+9$ .

Question 69

Find the function $p(x)=f(x) \cdot g(x)$ when $f(x)=3 x^{2}$ and $g(x)=2 x+7$ .