Q: Solve for $x$ in the interval $0 \leq x < 2\pi$: $2\sin x = 1$.

$x = \frac{\pi}{6}, \frac{5\pi}{6}$

Question 1

Let $f(x) = \sqrt{x-2}$ and $g(x) = \ln(x+1)$. What is the domain of $f(g(x))$?

Accepted Answer

$x \geq e^2 - 1$

Question 2

Which of the following is the inverse of the function $f(x) = 3x + 2$?

Accepted Answer

$f^{-1}(x) = \frac{x-2}{3}$

Question 3

Suppose the population of a bacteria culture is modeled by $A(t) = 500e^{0.3t}$, where $t$ is in hours. How long will it take for the population to reach 2000?

Accepted Answer

$t = \frac{\ln 4}{0.3}$

Question 4

Which transformation must be applied to the graph of $f(x) = x^2$ to obtain the graph of $g(x) = -(x-3)^2 + 2$?

Accepted Answer

Shift right 3 units, reflect over the $x$-axis, shift up 2 units

Question 5

Solve for $x$: $\log_2(x+1) = 3$.

Question 6

A radioactive substance decays according to $A(t) = A_0 e^{-0.05t}$. If the initial amount is 100 grams, how much remains after 20 years?

Accepted Answer

$A(20) = 100e^{-1} \approx 36.79$ grams

Question 7

Find the amplitude and period of $y = 3\sin(2x)$.

Accepted Answer

Amplitude: $3$, Period: $\pi$

Question 8

If $\sin 	heta = \frac{3}{5}$ and $	heta$ is in Quadrant II, what is $\cos 	heta$?

Accepted Answer

$\cos 	heta = -\frac{4}{5}$

Question 9

Which of the following is equivalent to $\log_3 81$?

Accepted Answer

$4$

Question 10

A Ferris wheel has a radius of 20 meters and rotates at a rate of 1 revolution every 2 minutes. What is the linear speed (in meters per minute) of a seat on the wheel?

Accepted Answer

$40\pi$ meters/minute

Question 11

Solve for $x$: $2^{x+1} = 16$.

Accepted Answer

$x = 3$

Question 12

Which of the following is the correct sum-to-product identity for $\sin A + \sin B$?

Accepted Answer

$2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)$

Question 13

If $	an 	heta = -\sqrt{3}$ and $	heta$ is in Quadrant IV, what is $	heta$?

Accepted Answer

$330^\circ$

Question 14

Which of the following is the correct double-angle formula for $\sin 2	heta$?

Accepted Answer

$2\sin	heta\cos	heta$

Question 15

Given $\cos(2x) = 1 - 2\sin^2 x$, what is $\sin^2 x$ in terms of $\cos(2x)$?

Accepted Answer

$\sin^2 x = \frac{1 - \cos(2x)}{2}$

Question 16

If $\sin \alpha = \frac{5}{13}$ and $\alpha$ is in Quadrant II, what is $\cos \alpha$?

Accepted Answer

$-\frac{12}{13}$

Question 17

Which of the following is the correct expression for $\log_b(MN)$ using properties of logarithms?

Accepted Answer

$\log_b M + \log_b N$

Question 18

A point $(x, y)$ lies on the unit circle and corresponds to an angle $	heta = \frac{2\pi}{3}$. What are the coordinates of the point?

Accepted Answer

$\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$

Question 19

Solve for $x$ in the interval $0 \leq x < 2\pi$: $2\sin x = 1$.

Accepted Answer

$x = \frac{\pi}{6}, \frac{5\pi}{6}$

Precalculus Final Exam Practice: Functions, Exponentials, Logarithms, and Trigonometry

Study Guide - Practice Questions

Study Guide - Flashcards