Q: Given $f(x) = x^2$ and $g(x) = 2x + 1$, find $g(f(2))$.

$g(f(2)) = g(4) = 2 \cdot 4 + 1 = 9$

Q: Which of the following statements is TRUE about the inverse of a function?

The inverse function reverses the roles of the input and output.

Q: If $f(x) = x^3$ and $g(x) = \sqrt[3]{x}$, what is $(f \circ g)(x)$?

$(f \circ g)(x) = f(g(x)) = (\sqrt[3]{x})^3 = x$

Q: A function $f$ is invertible if and only if:

$f$ is one-to-one (passes the horizontal line test)

Question 1

Given the function $f(x) = 2x + 1$ and $g(x) = x^2 - 3$, what is the value of $(f \circ g)(2)$?

Accepted Answer

$f(g(2)) = 2(2^2 - 3) + 1 = 2(1) + 1 = 3$

Question 2

Which of the following intervals describes where the function $h(x)$ is decreasing, given the graph below?

Accepted Answer

$(2, 4)$

Question 3

If $f(x) = 3x - 2$ and $g(x) = x + 4$, what is $(f + g)(x)$?

Accepted Answer

$3x - 2 + x + 4 = 4x + 2$

Question 4

Given $f(x) = x^2$ and $g(x) = x - 1$, find $(f - g)(3)$.

Accepted Answer

$f(3) - g(3) = 9 - 2 = 7$

Question 5

A company’s monthly cost to produce $x$ units is $C(x) = 2500 + 1.25x$. If each unit sells for $3.00$, write the revenue function $R(x)$ and the profit function $P(x)$.

Accepted Answer

$R(x) = 3x$, $P(x) = 3x - (2500 + 1.25x)$

Question 6

Given $f(x) = x^2 + 1$ and $g(x) = 2x$, find $(f \cdot g)(-1)$.

Accepted Answer

$f(-1) \cdot g(-1) = (1 + 1) \cdot (-2) = 2 \cdot (-2) = -4$

Question 7

If $f(x) = 2x + 3$ and $g(x) = x - 1$, what is the domain of $\left(\frac{f}{g}\right)(x)$?

Accepted Answer

$x 
eq 1$

Question 8

Given $f(x) = 3x + 4$, find the inverse function $f^{-1}(x)$.

Accepted Answer

$f^{-1}(x) = \frac{x - 4}{3}$

Question 9

Which of the following graphs represents a one-to-one function?

Accepted Answer

A graph that passes both the vertical and horizontal line tests

Question 10

Suppose $f(x) = x^2$ and $g(x) = x + 1$. What is $(f \circ g)(-2)$?

Accepted Answer

$f(g(-2)) = f(-1) = (-1)^2 = 1$

Question 11

Given the piecewise function $f(x) = \begin{cases} 2x + 3 & x < -2 \ -x + 2 & x \geq -2 \end{cases}$, what is $f(-3)$?

Accepted Answer

$2(-3) + 3 = -6 + 3 = -3$

Question 12

If $f(x) = \frac{2x+1}{3x-4}$, what is the domain of $f(x)$?

Accepted Answer

$x 
eq \frac{4}{3}$

Question 13

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

Accepted Answer

$g(f(2)) = g(4) = 2 \cdot 4 + 1 = 9$

Question 14

Which of the following statements is TRUE about the inverse of a function?

Accepted Answer

The inverse function reverses the roles of the input and output.

Question 15

Given the table below for $f(x)$:

$x$: $0, 1, 2, 3$

$f(x)$: $2, 4, 8, 16$

What is $f^{-1}(8)$?

Accepted Answer

$2$

Question 16

If $f(x) = x^3$ and $g(x) = \sqrt[3]{x}$, what is $(f \circ g)(x)$?

Accepted Answer

$(f \circ g)(x) = f(g(x)) = (\sqrt[3]{x})^3 = x$

Question 17

A function $f$ is invertible if and only if:

Accepted Answer

$f$ is one-to-one (passes the horizontal line test)

Question 18

Given $f(x) = 2x - 5$, find $x$ such that $f(x) = 7$.

Accepted Answer

$x = 6$

Question 19

Suppose $f(x)$ is increasing on $(-\infty, 2)$ and decreasing on $(2, \infty)$. What is the nature of $x = 2$?

Accepted Answer

$x = 2$ is a local maximum

Question 20

If $f(x) = x^2$ and $g(x) = x + 1$, which of the following is the correct expression for $(f + g)(x)$?

Accepted Answer

$x^2 + x + 1$

A5-A7

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