$$\tan \theta = \frac{\sin \theta}{\cos \theta}$$ and $$\sec \theta = \frac{1}{\cos \theta}$$, so $$\frac{\tan \theta}{\sec \theta} = \sin \theta$$

Question 1

Which of the following correctly expresses $$a^{-m}$$ using only positive exponents?

Accepted Answer

$$a^{-m}$$ = \frac{1}{$$a^{m}$$}$$

Question 2

A biologist models the population of bacteria as $$P(t) = 4 \cdot 2$$^{t}$$, where t$ is time in hours. What is the population after 3 hours?

Accepted Answer

$$P(3) = 32$$

Question 3

Which of the following is the correct domain and range for the circle $$(x+2)^2$$ + $$(y-5)^2 = 16$?

Accepted Answer

Domain: $$[-6, 2]$$, Range: $$[1, 9]$$

Question 4

A sound engineer uses the logarithmic property to expand $$\log_2$ \left(\frac{6x}{y}\right). $$Which is the correct expansion?

Accepted Answer

$$\log_2$ 6 + \log_2$ x - \log_2$ y$$

Question 5

Which of the following is the correct solution to $$x^2 + 3x - 28 = 0$ by factoring?

Accepted Answer

$$x = -7, 4$$

Question 6

A pharmaceutical company needs to solve $$2x^2 = 3x + 9$ for x$ using the quadratic formula. What are the solutions?

Accepted Answer

$$x = 3 \pm \sqrt{15} / 2$$

Question 7

Which of the following is the correct fundamental trigonometric identity?

Accepted Answer

$$\sin^2$ x + \cos^2$ x = 1$$

Question 8

A pilot needs to find the exact value of $$\sin \frac{\pi}{4}$$ for navigation. What is the value?

Accepted Answer

$$\frac{\sqrt{2}}{2}$$

Question 9

Which of the following is the correct inverse function for $$f(x) = 4x - 3$$?

Accepted Answer

$$f^{-1}(x$$) = \frac{x+3}{4}$$

Question 10

A physicist needs to simplify $$m^{2/3}$$ $$m^{5/3}$$ using exponent rules. What is the result?

Accepted Answer

$$m^{7/3}$$

Question 11

Which of the following is the correct solution to $$\sin 	heta = 0$$ for $$	heta in$$ $$[0, 2\pi]$$?

Accepted Answer

$$	heta = 0, \pi$$

Question 12

A medical researcher uses the difference quotient for $$f(x) = x^2$ - 4. $$What is the simplified difference quotient?

Accepted Answer

$$2x + h$$

Question 13

Which of the following is the correct expansion for $$\log_k \frac{pq^2$}{m}$$?

Accepted Answer

$$\log_k p + 2 \log_k q - \log_k m$$

Question 14

A student is asked to solve $$	an x + \sqrt{3} = 0$$ for $$x in$$ $$[0, 2\pi]. $$What are the solutions?

Accepted Answer

$$x = \frac{2\pi}{3}, \frac{5\pi}{3}$$

Question 15

Which of the following is the correct range for $$\arcsin x$$?

Accepted Answer

$$[-\frac{\pi}{2}, \frac{\pi}{2}]$$

Question 16

A civil engineer needs to find the vertical asymptote for $$f(x) = \frac{x-5}{x+3}. $$What is the vertical asymptote?

Accepted Answer

$$x = -3$$

Question 17

Which of the following is the correct solution to $$\sqrt{3} \csc x - 2 = 0$$ for $$x in$$ $$[0, 2\pi]$$?

Accepted Answer

$$x = \frac{\pi}{3}, \frac{2\pi}{3}$$

Question 18

A student is asked to verify the identity $$	an 	heta / \sec 	heta = \sin 	heta. $$Which of the following is the correct verification?

Accepted Answer

$$	an 	heta = \frac{\sin 	heta}{\cos 	heta}$$ and $$\sec 	heta = \frac{1}{\cos 	heta}$$, so $$\frac{	an 	heta}{\sec 	heta} = \sin 	heta$$

Question 19

Which of the following is the correct solution to $$x^2 = -16$ using the square root method?

Accepted Answer

$$x = -4i, 4i$$

Precalculus Final Review: Step-by-Step Study Guidance

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