Question 1

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

Accepted Answer

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)  $\frac{x+5}{(x-4)^{5 / 2}}$ 
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)  $\frac{x+5}{(x-4)^{5 / 2}}$ D

Question 2

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

Accepted Answer

$p(x)=6 x^{3}+21 x^{2}$

Question 3

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

Accepted Answer

$p(x)=3 x-5$

Question 4

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

Accepted Answer

The answer of If $h(x)=f(g(x))$, what is the value of...

Question 5

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

Accepted Answer

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

Question 6

The following table gives 3 functions, $f, g$, and $h$.
 
Which of the following statements is true?

Accepted Answer

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$ 
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 7

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

Accepted Answer

A) True 
 B)False

Question 8

Is the function graphed below invertible?

Accepted Answer

A) True 
 B)False

Question 9

The following figure defines a function $f$.
 
Which of the following quantities is greater?

Accepted Answer

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

Question 10

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

Accepted Answer

A) True 
 B)False

Question 11

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

Accepted Answer

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

Question 12

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

Accepted Answer

A) True 
 B)False

Question 13

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

Accepted Answer

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$ 
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 14

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".

Accepted Answer

The answer of Given the graph of $f$ below, solve...

Question 15

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

Accepted Answer

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

Question 16

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

Accepted Answer

The answer of Find $f(f(0))$ for \(f(x)=\left\{\begin{array}{cc}0 & x \leq...

Question 17

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

Accepted Answer

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

Question 18

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 $t$.

Accepted Answer

The answer of Let $f(t)$ be the number of men...

Question 19

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 $m(n(-3))$.

Accepted Answer

The answer of The functions $m(x)$ and $n(x)$ are defined...

Question 20

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 $h(x)=f(x) * g(x)$.

Accepted Answer

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. 
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.

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

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

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

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

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

The following table gives 3 functions, $f, g$ , and $h$ . Which of the following statements is true?

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

Is the function graphed below invertible?

The following figure defines a function $f$ . Which of the following quantities is greater?

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

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

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

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

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".

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

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

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

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 $m(n(-3))$ .

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 $h(x)=f(x) * g(x)$ .

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Exam 10: Compositions, Inverses, and Combinations of Functions

Let f(x)=x+5,g(x)=x5f(x)=x+5, g(x)=x^{5}f(x)=x+5,g(x)=x5 , and h(x)=x−4h(x)=\sqrt{x-4}h(x)=x−4​ . What is f(x)g(h(x))\frac{f(x)}{g(h(x))}g(h(x))f(x)​ ?

Find the function p(x)=f(x)⋅g(x)p(x)=f(x) \cdot g(x)p(x)=f(x)⋅g(x) when f(x)=3x2f(x)=3 x^{2}f(x)=3x2 and g(x)=2x+7g(x)=2 x+7g(x)=2x+7 .

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

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

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

The following table gives 3 functions, f,gf, gf,g , and hhh . Which of the following statements is true?

Is the function y=∣4x∣y=\left|\frac{4}{x}\right|y=​x4​​ invertible if the domain of fff is all real numbers?

Is the function graphed below invertible?

The following figure defines a function fff . Which of the following quantities is greater?

If f(x)=x2+2xf(x)=x^{2}+2 xf(x)=x2+2x and g(x)=2−xg(x)=2-xg(x)=2−x , does f(g(x))=x2−2x+4f(g(x))=x^{2}-2 x+4f(g(x))=x2−2x+4 ?

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

Let f(x)=x2+3f(x)=x^{2}+3f(x)=x2+3 , and g(x)=1x+1 g(x)=\frac{1}{x}+1g(x)=x1​+1 . Does g(f(x))=1x2+1x2+3?g(f(x))=\frac{1 x^{2}+1}{x^{2}+3} ?g(f(x))=x2+31x2+1​?

If H(x)=f(g(x))=e4x+2H(x)=f(g(x))=e^{4 x+2}H(x)=f(g(x))=e4x+2 , which of the following could be true? Mark all that apply.

Given the graph of fff below, solve f(x)=xf(x)=xf(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".

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

Find f(f(0))f(f(0))f(f(0)) for f(x)={0x≤3x3<x<88x≥8f(x)=\left\{\begin{array}{cc}0 & x \leq 3 \\ x & 3<x<8 \\ 8 & x \geq 8\end{array}\right.f(x)=⎩⎨⎧​0x8​x≤33<x<8x≥8​

Let f(x)=x+1f(x)=x+1f(x)=x+1 and g(x)=x2g(x)=x^{2}g(x)=x2 . What is f(x)g(x)f(x) g(x)f(x)g(x) ?

The functions m(x)m(x)m(x) and n(x)n(x)n(x) are defined by the graph below. The dashed graph is m(x)m(x)m(x) and the solid graph is n(x)n(x)n(x) . Evaluate m(n(−3))m(n(-3))m(n(−3)) .

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

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Polynomial and Rational Functions

Vectors and Matrices

Sequences and Series

Parametric Equations and Conic Sections

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

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

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

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

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

The following table gives 3 functions, $f, g$ , and $h$ . Which of the following statements is true?

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

The following figure defines a function $f$ . Which of the following quantities is greater?

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

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

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

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

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".

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

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

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

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 $m(n(-3))$ .

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 $h(x)=f(x) * g(x)$ .