$(-\infty, -3) \cup (-3, 1) \cup (1, \infty)$

A relation where an input value is assigned to two different output values.

Question 1

Which of the following best describes a function?

Accepted Answer

A rule that assigns to each input value exactly one output value.

Question 2

Given the function $f(x) = \sqrt{9 - x}$, what is the domain of $f(x)$?

Accepted Answer

$(-\infty, 9]$

Question 3

Which of the following graphs passes the vertical line test and represents a function?

Accepted Answer

A graph where every vertical line crosses the curve at most once.

Question 4

If $f(x) = 2x - 3$, $g(t) = 2t - 3$, and $C(n) = 2n - 3$, which of the following is true?

Accepted Answer

All three represent the same function rule.

Question 5

Let $h(x) = \ln(x-2)$. What is the domain of $h(x)$?

Accepted Answer

$(2, \infty)$

Question 6

Suppose $f(x) = 3x + 1$. What is $f(x+h)$?

Accepted Answer

$3x + 3h + 1$

Question 7

Given $f(x) = x^2 + 3$ and $g(x) = 4x - 1$, what is $(f+g)(x)$?

Accepted Answer

$x^2 + 4x + 2$

Question 8

If $f(x) = x^2 + 3$, $g(x) = 4x - 1$, and $h(x) = e^x$, what is $y = \frac{f(x)}{g(x)} - 3h(x)$?

Accepted Answer

$\frac{x^2 + 3}{4x - 1} - 3e^x$

Question 9

What is the difference quotient for $f(x) = x - 2x^2$?

Accepted Answer

$1 - 4x - 2h$

Question 10

Let $f(x) = x - 5$ and $g(x) = 0.75x$. If you use the coupon first and then the store discount, what is the resulting function $h(x)$?

Accepted Answer

$h(x) = 0.75(x - 5)$

Question 11

Given the piecewise function $f(x) = \begin{cases} x+6 & -3 \leq x < 0 \ -x^2+5 & 0 \leq x < 2 \ 3 & 2 \leq x \leq 5 \end{cases}$, what is $f(1)$?

Accepted Answer

$4$

Question 12

For the function $f(x) = \frac{1}{(x+3)(x-1)}$, what is the domain?

Accepted Answer

$(-\infty, -3) \cup (-3, 1) \cup (1, \infty)$

Question 13

If $f(x) = 2x - 1$ and $g(x) = e^x$, what is $(f \circ g)(x)$?

Accepted Answer

$2e^x - 1$

Question 14

Which of the following is NOT a function?

Accepted Answer

A relation where an input value is assigned to two different output values.

Question 15

Given $f(x) = x^2 + 3$ and $g(x) = 4x - 1$, what is $(f \circ g)(x)$?

Accepted Answer

$(4x - 1)^2 + 3$

Question 16

If $y = e^{2x}$, decompose this as a composition of two functions $f$ and $g$ such that $y = f(g(x))$.

Accepted Answer

$f(u) = e^u$, $g(x) = 2x$

Question 17

A medical test result is modeled by $T(x) = \sqrt{1 + x^2}$, where $x$ is a measured value. What is the domain of $T(x)$?

Accepted Answer

$(-\infty, \infty)$

Question 18

A chemical reaction rate is given by $R(t) = \ln(t-2)$, where $t$ is time in seconds. For which values of $t$ is the reaction rate defined?

Accepted Answer

$t > 2$

Study Guide: Basics of Functions for Calculus

Study Guide - Practice Questions