Q: A function $$f(x) = \frac{1}{\sqrt{x-1}} is $$defined for which values of $$x$$?

$$x \u003e 1$$ \text{ and } x \neq 1$

Q: Which of the following is the correct general solution to $$\tan 5\theta = -1$$?

$$\theta = -\frac{\pi}{20} + k\frac{\pi}{5}$$

Q: Which of the following is the correct domain for the function $$g(x) = \frac{1}{|x| - 2}$$?

$$x \in \mathbb{R}, x \neq 2, x \neq -2$$

Q: Which of the following is the correct expression for $$\sin^4$ \theta in $$terms of cosine functions (without even powers)?

$$\sin^4$ \theta = \frac{3}{8} - \frac{1}{2} \cos(2\theta) + \frac{1}{8} \cos(4\theta)$$

Q: Which of the following is the correct formula for converting an angle $$\alpha$$ from degrees to radians?

$$\alpha_{\text{rad}} = \alpha_{\text{deg}} \cdot \frac{\pi}{180}$$

Q: Which of the following is the correct expression for $$a \cos x + b \sin x in $$the form $$R \cos(x - \theta)$$?

$$R = \sqrt{a^2$ + b^2$}, \theta = \arctan\left(\frac{b}{a}\right)$$

Question 1

Which of the following correctly describes the set of all real numbers $$x$$ that satisfy $$|2x + 4| \u003c 1$$?

Accepted Answer

$$-5 \u003c x \u003c -3$$

Question 2

A function $$f(x) = \frac{1}{\sqrt{x-1}} is $$defined for which values of $$x$$?

Accepted Answer

$$x \u003e 1$$ 	ext{ and } x 
eq 1$

Question 3

Which of the following is the correct general solution to $$	an 5	heta = -1$$?

Accepted Answer

$$	heta = -\frac{\pi}{20} + k\frac{\pi}{5}$$

Question 4

A car wheel has a diameter of $$60 cm $$and the car moves at $$80 km/h. $$How many revolutions per minute does the wheel make?

Accepted Answer

$$425.5$$

Question 5

Which of the following is the correct binomial expansion for $$(a + b)^4$?

Accepted Answer

$$a^4$$ + $$4a^3b$$ + $$6a^2b^2$ + 4ab^3$ + b^4$

Question 6

Given $$f(x) = x^2$ + 2x^4$ + 1$$, which of the following statements is true?

Accepted Answer

$$f is an $$even function

Question 7

If $$a$$ and $$b$$ are real numbers, which property of absolute value is always true?

Accepted Answer

$$|ab| = |a||b|$$

Question 8

Which of the following is the correct formula for the area $$A of a $$sector with radius $$r$$ and angle $$\alpha (in $$radians)?

Accepted Answer

$$A = \frac{1}{2} r^2$ \alpha$$

Question 9

Which of the following is the correct domain for the function $$g(x) = \frac{1}{|x| - 2}$$?

Accepted Answer

$$x \in \mathbb{R}, x 
eq 2, x 
eq -2$$

Question 10

Which of the following is the correct expression for $$\sin^4$ 	heta in $$terms of cosine functions (without even powers)?

Accepted Answer

$$\sin^4$ 	heta = \frac{3}{8} - \frac{1}{2} \cos(2	heta) + \frac{1}{8} \cos(4	heta)$$

Question 11

Which of the following is the correct formula for the sum $$S of $$the first $$n$$ positive integers?

Accepted Answer

$$S = \frac{n(n+1)}{2}$$

Question 12

Which of the following is the correct solution set for the inequality $$4x + \frac{3}{x-2} \leq 1$$?

Accepted Answer

$$(-\infty,0) \cup [1,2) \cup [8,\infty)$$

Question 13

Which of the following is the correct formula for converting an angle $$\alpha$$ from degrees to radians?

Accepted Answer

$$\alpha_{	ext{rad}} = \alpha_{	ext{deg}} \cdot \frac{\pi}{180}$$

Question 14

Which of the following is the correct expression for $$a \cos x + b \sin x in $$the form $$R \cos(x - 	heta)$$?

Accepted Answer

$$R = \sqrt{a^2$ + b^2$}, 	heta = \arctan\left(\frac{b}{a}ight)$$

Question 15

Which of the following is the correct domain for the composite function $$f \circ g$$ where $$f(x) = \sqrt{x}$$ and $$g(x) = x^2$ - 4$$?

Accepted Answer

$$(-\infty, -2] \cup [2, \infty)$$

Question 16

Which of the following is the correct formula for the binomial coefficient $$\binom{n}{k}$$?

Accepted Answer

$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$

Question 17

Which of the following is the correct canonical form for the ellipse $$x^2 + 2x$$ + $$2y^2 - 8y + 7 = 0$ after completing the square?

Accepted Answer

$$\frac{(x+1)^2$}{2} + (y-2)^2$ = 1$$

Question 18

Which of the following is the correct general solution for $$\sin 	heta = x$$ where $$x \in [-1,1]$$?

Accepted Answer

$$	heta = \arcsin x + 2k\pi or$$ $$	heta = \pi - \arcsin x + 2k\pi$$,\ k \in \mathbb{Z}$

College Algebra: Core Concepts, Functions, Trigonometry, Induction, and Conic Sections

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