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

Accepted Answer

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 $$0 \leq x \u003c 2\pi$$: $$2\sin x = 1.$$

Accepted Answer

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

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

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