Q: Given $f(x) = x^2 - 4$ and $g(x) = x + 2$, find $(f+g)(x)$ and its domain.

$(f+g)(x) = x^2 + x - 2$, domain: $(-\infty, \infty)$

Q: Given $f(x) = \sqrt{x-1}$ and $g(x) = x^2$, find $(f \circ g)(x)$ and its domain.

$(f \circ g)(x) = \sqrt{x^2 - 1}$, domain: $(-\infty, -1] \cup [1, \infty)$

Q: A company’s profit $P$ (in dollars) is modeled by $P(x) = -2x^2 + 40x - 100$, where $x$ is the number of units sold (in hundreds). What is the maximum profit, and for how many units sold does it occur?

Maximum profit: $300$, at $x = 10$ (i.e., 1000 units)

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

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

Question 1

Given the points $A(2, -3)$ and $B(-4, 5)$, what is the distance between them?

Accepted Answer

$\sqrt{(2 - (-4))^2 + ((-3) - 5)^2} = \sqrt{36 + 64} = \sqrt{100} = 10$

Question 2

Find the midpoint of the segment joining the points $(-6, 8)$ and $(4, -2)$.

Accepted Answer

$\left( \frac{-6 + 4}{2}, \frac{8 + (-2)}{2} \right) = (-1, 3)$

Question 3

What is the standard equation of a circle with center $(3, -2)$ and radius $5$?

Accepted Answer

$(x - 3)^2 + (y + 2)^2 = 25$

Question 4

Given the equation of a circle $x^2 + y^2 - 6x + 4y + 9 = 0$, what is the center and radius?

Accepted Answer

Center: $(3, -2)$, Radius: $2$

Question 5

Which of the following relations is NOT a function?

Accepted Answer

$\{(1,2), (2,3), (1,4)\}$

Question 6

If $f(x) = 2x^2 - 3x + 1$, what is $f(-2)$?

Accepted Answer

$2(-2)^2 - 3(-2) + 1 = 8 + 6 + 1 = 15$

Question 7

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

Accepted Answer

$(-\infty, 4) \cup (4, \infty)$

Question 8

Find the slope of the line passing through the points $(7, -1)$ and $(2, 4)$.

Accepted Answer

$m = \frac{-1 - 4}{7 - 2} = \frac{-5}{5} = -1$

Question 9

Given the linear equation $3x - 2y = 6$, what is the slope and y-intercept?

Question 10

Write the equation of a line with slope $-2$ and y-intercept $5$.

Accepted Answer

$y = -2x + 5$

Question 11

Find the equation of the line passing through $(1, 2)$ and $(3, 6)$.

Accepted Answer

$y = 2x$

Question 12

Write the equation of a line passing through $(2, -1)$ and parallel to $y = 3x + 4$.

Accepted Answer

$y = 3x - 7$

Question 13

Write the equation of a line passing through $(0, 5)$ and perpendicular to $y = -\frac{1}{2}x + 3$.

Accepted Answer

$y = 2x + 5$

Question 14

A rectangular garden has a perimeter of $60$ meters. If the length is $5$ meters more than the width, what are the dimensions of the garden?

Accepted Answer

Length: $17.5$ m, Width: $12.5$ m

Question 15

Solve the inequality $2x - 5 < 7$ and write the solution in interval notation.

Accepted Answer

$(-\infty, 6)$

Question 16

Given $f(x) = x^2 - 4$ and $g(x) = x + 2$, find $(f+g)(x)$ and its domain.

Accepted Answer

$(f+g)(x) = x^2 + x - 2$, domain: $(-\infty, \infty)$

Question 17

Given $f(x) = \sqrt{x-1}$ and $g(x) = x^2$, find $(f \circ g)(x)$ and its domain.

Accepted Answer

$(f \circ g)(x) = \sqrt{x^2 - 1}$, domain: $(-\infty, -1] \cup [1, \infty)$

Question 18

A company’s profit $P$ (in dollars) is modeled by $P(x) = -2x^2 + 40x - 100$, where $x$ is the number of units sold (in hundreds). What is the maximum profit, and for how many units sold does it occur?

Accepted Answer

Maximum profit: $300$, at $x = 10$ (i.e., 1000 units)

Question 19

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

Accepted Answer

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

Question 20

Given $f(x) = x^2 - 1$ and $g(x) = x - 1$, find $\left(\frac{f}{g}\right)(x)$ and its domain.

Accepted Answer

$\frac{x^2 - 1}{x - 1}$, domain: $(-\infty, 1) \cup (1, \infty)$

College Algebra Exam 1 Study Guide: Graphs, Functions, and Linear Equations

Study Guide - Practice Questions

Study Guide - Flashcards