Q: Given $f(x) = x^2 + 3x - 4$ and $g(x) = 5x - 3$, what is $(f+g)(x)$ and its domain?

$(f+g)(x) = x^2 + 8x - 7$, \text{ domain: } (-\infty, \infty)$

Q: Given $f(x) = x^2 + 3x - 4$ and $g(x) = 5x - 3$, what is $\left(\frac{f}{g}\right)(x)$ and its domain?

$\frac{x^2 + 3x - 4}{5x - 3}$, \text{ domain: } x \neq \frac{3}{5}$

Q: Given $f(x) = \frac{1}{x}$ and $g(x) = \frac{4x}{x+1}$, find $(f \circ g)(x)$ and its domain.

$(f \circ g)(x) = \frac{x+1}{4x}$, \text{ domain: } x \neq 0, x \neq -1$

Q: Express $f(x) = x^2 + 4x - 1$ in the form $f(x) = a(x-h)^2 + k$. What is the vertex?

$f(x) = (x+2)^2 - 5$, \text{ vertex: } (-2, -5)$

Q: State the end behavior of the polynomial $P(x) = -2x^4 + 3x^2 - 2x + 4$.

As $x \to \infty$, $P(x) \to -\infty$; as $x \to -\infty$, $P(x) \to -\infty$

Q: A company models its profit $P(x)$ in thousands of dollars by $P(x) = -2x^2 + 8x + 10$, 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 is $26$ thousand dollars at $x = 2$ (i.e., 200 units)

Question 1

Given the piecewise function

$ f(x) = \begin{cases} 2x & 	ext{if } x < -1 \ -2x + 3 & 	ext{if } -1 \leq x \leq 3 \ \sqrt{x} & 	ext{if } x > 3 \end{cases} $

What is the value of $f(0)$?

Accepted Answer

$f(0) = 3$

Question 2

Given $f(x) = x^2 + 3x - 4$ and $g(x) = 5x - 3$, what is $(f+g)(x)$ and its domain?

Accepted Answer

$(f+g)(x) = x^2 + 8x - 7$, 	ext{ domain: } (-\infty, \infty)$

Question 3

Given $f(x) = x^2 + 3x - 4$ and $g(x) = 5x - 3$, what is $\left(\frac{f}{g}\right)(x)$ and its domain?

Accepted Answer

$\frac{x^2 + 3x - 4}{5x - 3}$, 	ext{ domain: } x 
eq \frac{3}{5}$

Question 4

Given $f(x) = \frac{1}{x}$ and $g(x) = \frac{4x}{x+1}$, find $(f \circ g)(x)$ and its domain.

Accepted Answer

$(f \circ g)(x) = \frac{x+1}{4x}$, 	ext{ domain: } x 
eq 0, x 
eq -1$

Question 5

State the sequence of transformations that takes $f(x) = x^2$ to $g(x) = -2(x+4)^2 + 4$.

Accepted Answer

Shift left 4 units, vertical stretch by 2, reflect over x-axis, shift up 4 units

Question 6

Find the equation of the graph obtained by taking $y = x^2$, stretching it vertically by a factor of 3, shifting up 3 units, and left 4 units.

Accepted Answer

$y = 3(x+4)^2 + 3$

Question 7

Find $(2 - 7i) + (8 + 4i)$.

Accepted Answer

$10 - 3i$

Question 8

Find $(3 - 5i)(8 - 2i)$.

Accepted Answer

$34 - 46i$

Question 9

Solve the equation $x^2 + 8x = 9$ by completing the square.

Accepted Answer

$x = -4 \pm \sqrt{25}$

Question 10

Solve the equation $3x^2 + x = 5$ using the quadratic formula.

Accepted Answer

$x = \frac{-1 \pm \sqrt{61}}{6}$

Question 11

Express $f(x) = x^2 + 4x - 1$ in the form $f(x) = a(x-h)^2 + k$. What is the vertex?

Accepted Answer

$f(x) = (x+2)^2 - 5$, 	ext{ vertex: } (-2, -5)$

Question 12

Solve the equation $\frac{2}{x+5} + \frac{1}{x-5} = \frac{16}{x^2-25}$.

Accepted Answer

$x = 3$

Question 13

Solve the equation $\sqrt{x+1} + 1 = x - 1$.

Accepted Answer

$x = 3$

Question 14

Solve the inequality $|2x + 5| < 9$. Express your answer in interval notation.

Accepted Answer

$(-7, 2)$

Question 15

State the end behavior of the polynomial $P(x) = -2x^4 + 3x^2 - 2x + 4$.

Accepted Answer

As $x 	o \infty$, $P(x) 	o -\infty$; as $x 	o -\infty$, $P(x) 	o -\infty$

Question 16

Solve the inequality $\frac{4x+1}{x+2} \leq 3$. Express your answer in interval notation.

Accepted Answer

$(-\infty, -3] \cup (-2, 1]$

Question 17

A company models its profit $P(x)$ in thousands of dollars by $P(x) = -2x^2 + 8x + 10$, 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 is $26$ thousand dollars at $x = 2$ (i.e., 200 units)

Question 18

Which of the following describes the domain of the function $f(x) = \sqrt{5 - x}$?

Accepted Answer

$(-\infty, 5]$

Question 19

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

Accepted Answer

$9$

College Algebra Review: Functions, Transformations, Equations, and Inequalities

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