Q: Given $f(x) = \frac{3}{x-1}$ and $g(x) = \frac{2x-1}{x}$, find $f(g(x))$ and state its domain.

$f(g(x)) = \frac{3x}{x-2}$,\ x \neq 0,\ x \neq 2$

Q: Given $f(x) = \frac{2x+1}{1-x}$, state the domain and range of the inverse function.

Domain: $x \neq 1$, Range: $y \neq -1$

Question 1

Add the following complex numbers: $(-5 + 7i) + (-8 - 11i)$

Accepted Answer

$-13 - 4i$

Question 2

Find the product: $(5 + 3i)(2 - i)$

Accepted Answer

$13 + i$

Question 3

Divide and express the result in standard form: $\frac{3}{2 - i}$

Accepted Answer

$\frac{6 + 3i}{5}$

Question 4

Solve the quadratic equation using the quadratic formula: $2x^2 + 2x + 3 = 0$

Question 5

Divide using long division: $\frac{x^4 - x^3 - 2x^2 + 2x + 3}{x^2 + 1}$

Accepted Answer

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

Question 6

Given $f(x) = \frac{3}{x-1}$ and $g(x) = \frac{2x-1}{x}$, find $f(g(x))$ and state its domain.

Accepted Answer

$f(g(x)) = \frac{3x}{x-2}$,\ x 
eq 0,\ x 
eq 2$

Question 7

State the domain and range of the relation $\{(-3, -1), (3, 4), (-5, -7), (5, -1)\}$ and indicate if the relation is one-to-one.

Accepted Answer

Domain: $\{-5, -3, 3, 5\}$, Range: $\{-7, -1, 4\}$, One-to-one

Question 8

Given $f(x) = \frac{2x+1}{1-x}$, state the domain and range of the inverse function.

Accepted Answer

Domain: $x 
eq 1$, Range: $y 
eq -1$

Question 9

Solve the inequality $x^2 - 7x^2 - 18x > 0$ algebraically and include a sign chart.

Accepted Answer

$x < -2$ or $x > 9$

Question 10

Solve the inequality $\frac{3x^2 + 10x - 8}{x-1} \geq 0$.

Accepted Answer

$(-\infty, -4] \cup [0.5, 1) \cup (1, \infty)$

Question 11

For the polynomial $f(x) = 9x^4 - 16x^3 + 60x^2 - 64x - 22$, which of the following is a possible rational root according to the Rational Root Theorem?

Accepted Answer

$\frac{11}{9}$

Question 12

Given that $-2i$ is a root of $f(x) = x^4 + 4x^3 + 24x^2 + 16x + 80$, find the remaining three solutions.

Accepted Answer

$2i, -2 + 4i, -2 - 4i$

Question 13

If $f(x) = x^2 - 4x + 6$ and $g(x) = x - 2$, find $(f \circ g)(3)$.

Accepted Answer

$f(g(3)) = 6$

Question 14

Given the function $f(x) = x^2 - 4x + 6$, determine if $f$ is one-to-one.

Accepted Answer

No, because it fails the horizontal line test.

Question 15

Which of the following best describes the domain of $f(x) = \frac{1}{x^2 - 4}$?

Accepted Answer

$x 
eq -2, 2$

Question 16

If $f(x) = x^2$ and $g(x) = \sqrt{x}$, what is the domain of $(g \circ f)(x)$?

Accepted Answer

$(-\infty, \infty)$

Question 17

Given $f(x) = x^2 - 4x + 6$, what is the vertex of the parabola?

Accepted Answer

$(2, 2)$

Question 18

If $f(x) = x^2 - 4x + 6$, what is the axis of symmetry?

Accepted Answer

$x = 2$

Question 19

Given $f(x) = x^2 - 4x + 6$, for which $x$ does $f(x) = 0$?

Accepted Answer

$x = 2 \pm \sqrt{2}$

Precalculus Exam 3 Review: Key Concepts and Practice Problems

Study Guide - Practice Questions

Study Guide - Flashcards