Q: Which rational function $$R(x)$$ has a graph that crosses the $$x-$$axis at $$-1$$, touches the $$x-$$axis at $$-4$$, has vertical asymptotes at $$x = 2$$ and $$x = 3$$, and has one horizontal asymptote at $$y = -2$$?

$$R(x) = \frac{-2(x + 1)(x + 4)^2$}{(x + 2)^2$(x - 3)},\ x \neq -2, 3$$

Q: Solve the equation $$2 \cos \theta + 3 = 2$$ for $$0 \leq \theta \u003c 2\pi.$$

$$\theta = \frac{2\pi}{3},\ \frac{4\pi}{3}$$

Q: Solve the equation $$\cos(2\theta) = \sqrt{2} - \cos(2\theta)$$ for $$0 \leq \theta \u003c 2\pi.$$

$$\theta = \frac{\pi}{8}, \frac{7\pi}{8}, \frac{9\pi}{8}, \frac{15\pi}{8}$$

Q: Which of the following is the correct solution set for $$4 \sin^2$ \theta = 1 on$$ $$0 \leq \theta \u003c 2\pi$$?

$$\theta = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6}$$

Question 1

Find the domain of the rational function $$R(x) = \frac{x^2$ - 3x^2$}{x^2$ + 7x - 30}.$$

Accepted Answer

$$\{x \mid x 
eq -10, 3\}$$

Question 2

Solve the equation $$\log(5 + x) - \log(x - 3) = \log 5$$ for $$x.$$

Accepted Answer

$$x = 5$$

Question 3

Find the domain of the function $$f(x) = \frac{x^2$ + 7}{x^2$ + 4x - 7}.$$

Accepted Answer

$$\{x \mid x 
eq -4, 7\}$$

Question 4

Solve the equation $$\log_3$ x + \log_3$ (x - 24) = 4$$ for $$x.$$

Accepted Answer

$$x = 27$$

Question 5

Find the $$y-$$intercept of the function $$f(x) = \frac{24x - 28}{(x^2$ + 20)(x - 6)}.$$

Accepted Answer

$$(0, -\frac{7}{30})$$

Question 6

Find the $$x-$$intercepts of $$f(x) = \frac{2x + 7}{x^2$ - 9}.$$

Accepted Answer

$$(-\frac{7}{2}, 0)$$

Question 7

Solve the equation $$\log_5$(x - 1) = 3 + \log_5$(x - 3)$$ for $$x.$$

Accepted Answer

$$x = 127$$

Question 8

Find the vertical asymptotes of the rational function $$h(x) = \frac{7}{x^2$ - 49}.$$

Accepted Answer

$$x = -7, x = 7$$

Question 9

Give the equation of the oblique asymptote, if any, of the function $$f(x) = \frac{2x^2$ + 11x + 8}{x^2$ + 6x + 5}.$$

Accepted Answer

$$y = 2x - 1$$

Question 10

The half-life of silicon-32 is 70 years. If 30 grams is present now, how much will be present in 200 years? Round your answer to three decimal places.

Accepted Answer

$$4.140$$ grams

Question 11

The half-life of plutonium-234 is 9 hours. If 80 milligrams are present now, how much will be present in 2 days? Round your answer to three decimal places.

Accepted Answer

$$1.984 mg$$

Question 12

During its first year of operation, 200,000 people visited Rave Amusement Park. Six years later, the number has grown to 400,000. Assuming uninhibited growth, find the exponential growth function of the model data.

Accepted Answer

$$f(t) = 200,000 e$$^{0.238t}$$

Question 13

Which rational function $$R(x)$$ has a graph that crosses the $$x-$$axis at $$-1$$, touches the $$x-$$axis at $$-4$$, has vertical asymptotes at $$x = 2$$ and $$x = 3$$, and has one horizontal asymptote at $$y = -2$$?

Accepted Answer

$$R(x) = \frac{-2(x + 1)(x + 4)^2$}{(x + 2)^2$(x - 3)},\ x 
eq -2, 3$$

Question 14

Solve the equation $$2 \cos 	heta + 3 = 2$$ for $$0 \leq 	heta \u003c 2\pi.$$

Accepted Answer

$$	heta = \frac{2\pi}{3},\ \frac{4\pi}{3}$$

Question 15

Simplify the expression $$\frac{\cos 	heta}{1 + \sin 	heta} + 	an 	heta.$$

Accepted Answer

$$\sec 	heta$$

Question 16

Establish the identity: $$\cos^2$ x - \sin^2$ x = 1 - 2\sin^2$ x.$$

Accepted Answer

True, by using the double angle identity for cosine

Question 17

Solve the equation $$\cos(2	heta) = \sqrt{2} - \cos(2	heta)$$ for $$0 \leq 	heta \u003c 2\pi.$$

Accepted Answer

$$	heta = \frac{\pi}{8}, \frac{7\pi}{8}, \frac{9\pi}{8}, \frac{15\pi}{8}$$

Question 18

Which of the following is the correct solution set for $$4 \sin^2$ 	heta = 1 on$$ $$0 \leq 	heta \u003c 2\pi$$?

Accepted Answer

$$	heta = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6}$$

Precalculus Review: Rational Functions, Logarithms, Exponential Models, and Trigonometry

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