Question 1

Given the relation $\{(2,3), (4,5), (8,8)\}$, determine whether it is a function. What are the domain and range?

Accepted Answer

It is a function. Domain: $\{2, 4, 8\}$, Range: $\{3, 5, 8\}$

Question 2

For the function $f(x) = x^2 - 13x + 4$, what is the domain?

Accepted Answer

$(-\infty, \infty)$

Question 3

Find the domain of $f(x) = \sqrt{x+5}$.

Accepted Answer

$[-5, \infty)$

Question 4

Find the domain of $f(x) = \frac{1}{\sqrt{x+11}}$.

Accepted Answer

$(-11, \infty)$

Question 5

Given $g(x) = \frac{\sqrt{x-2}}{x-9}$, what is the domain of $g$?

Accepted Answer

$[2, 9) \cup (9, \infty)$

Question 6

Given $f(x) = \frac{3x+10}{x^2 + 2x^2 - 9x - 18}$, find the domain of $f$.

Accepted Answer

$(-\infty, -3) \cup (-3, -2) \cup (-2, 3) \cup (3, \infty)$

Question 7

Suppose $f(x) = x^2 - 5x + 5$. Find and simplify the difference quotient $\frac{f(x+h) - f(x)}{h}$.

Accepted Answer

$2x + h - 5$

Question 8

Given $f(x) = 6x$ and $g(x) = x + 7$, find $(f \circ g)(x)$.

Accepted Answer

$6x + 42$

Question 9

Given $f(x) = 3x + 3$ and $g(x) = \frac{x-3}{3}$, are $f$ and $g$ inverses of each other?

Accepted Answer

Yes, $f$ and $g$ are inverses of each other.

Question 10

Which of the following equations defines $y$ as a function of $x$?

Accepted Answer

$x + y = 41$

Question 11

Given the function $f(x) = \sqrt{x-3} - 4$, what is the domain of $f$?

Accepted Answer

$[3, \infty)$

Question 12

Given the relation $\{(4,1), (4,5), (6,1), (6,5)\}$, is this a function? What is the domain?

Accepted Answer

No, domain is $\{4, 6\}$

Question 13

If $f(x) = x^2 + 6x + 7$, what is $f(x+7)$?

Accepted Answer

$x^2 + 20x + 98$

Question 14

Which of the following graphs passes the vertical line test and thus represents $y$ as a function of $x$?

Accepted Answer

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

Question 15

Given $f(x) = x^2 - 13x + 4$, what is the range of $f$?

Accepted Answer

$[k, \infty)$ where $k$ is the minimum value of $f$

Question 16

Given $f(x) = x^2 - 5x + 5$ and $g(x) = 3x^2$, what is the domain of $\frac{f}{g}$?

Accepted Answer

$(-\infty, 0) \cup (0, \infty)$

Question 17

Given $f(x) = x^2 - 13x + 4$, what is the $x$-value of the vertex?

Accepted Answer

$x = \frac{13}{2}$

Question 18

Given $f(x) = x^2 - 13x + 4$, what is the $y$-value of the vertex?

Accepted Answer

$y = 4 - 13 \cdot \frac{13}{2} + \left(\frac{13}{2}\right)^2$

Question 19

Given $f(x) = x^2 - 13x + 4$, for which values of $x$ is $f(x)$ increasing?

Accepted Answer

$\left(\frac{13}{2}, \infty\right)$

College Algebra Study Guide: Functions, Domains, Ranges, and Transformations

Study Guide - Practice Questions