Question 1

Which of the following is the least nonnegative coterminal angle in radians for $$	heta = -\frac{5\pi}{4}$$?

Accepted Answer

$$\frac{3\pi}{4}$$

Question 2

Given $$	an 	heta = -2$$ and $$	heta is in $$Quadrant II, what is the value of $$\sin 	heta$$?

Accepted Answer

$$\frac{2}{\sqrt{5}}$$

Question 3

If $$f(x) = \sqrt{x}$$, what is the simplified form of the difference quotient $$\frac{f(x+h) - f(x)}{h}$$?

Accepted Answer

$$\frac{1}{\sqrt{x+h} + \sqrt{x}}$$

Question 4

A ladder leans against a wall making a $$60^\circ$$ angle with the ground. If the foot of the ladder is 4 meters from the wall, how long is the ladder?

Accepted Answer

$$8$$ meters

Question 5

Expand $$\log_b(xy^2$)$$ using logarithmic properties.

Accepted Answer

$$\log_b x + 2\log_b y$$

Question 6

What is the domain of the function $$f(x) = \frac{1}{x^2$ - 4}$$?

Accepted Answer

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

Question 7

Given a sector of a circle with area $$A = 10$$ and radius $$r = 4$$, what is the central angle $$	heta in $$radians?

Accepted Answer

$$1.25$$

Question 8

Condense the expression $$2\log_3$ x - \frac{1}{2}\log_3$ y$$ into a single logarithm.

Accepted Answer

$$\log_3$ \left( \frac{x^2$}{\sqrt{y}} \right)$$

Question 9

If $$f(x) = 2(x-3)^2$ + 1$$, which transformation has been applied to the parent function $$y = x^2$?

Accepted Answer

Vertical stretch by 2, right shift by 3, up shift by 1

Question 10

Find the inverse of $$f(x) = x^2$ + 4$$ for $$x \geq 0.$$

Accepted Answer

$$f^{-1}(x$$) = \sqrt{x-4}$$

Question 11

A radioactive substance decays to $$\frac{1}{8} of $$its original amount in 24 hours. What is its half-life?

Accepted Answer

$$6$$ hours

Question 12

What is the reference angle for $$	heta = \frac{8\pi}{3}$$?

Accepted Answer

$$\frac{2\pi}{3}$$

Question 13

Simplify $$\sin(\frac{\pi}{2} - x)$$ using a cofunction identity.

Accepted Answer

$$\cos x$$

Question 14

What is the exact value of $$\csc(\frac{5\pi}{3})$$?

Accepted Answer

$$2\sqrt{3}/3$$

Question 15

Given $$\cos 	heta = -\frac{3}{5}$$ and $$	heta is in $$Quadrant III, find $$\sin 	heta.$$

Accepted Answer

$$-\frac{4}{5}$$

Question 16

Solve for $$x$$: $$\log_x 16 = 2.$$

Accepted Answer

$$4$$

Question 17

Solve for $$x$$: $$e^{2x} = 7$.

Accepted Answer

$$x = \frac{1}{2} \ln 7$$

Question 18

Which of the following rational functions has a removable discontinuity at $$x = 2$$?

Accepted Answer

$$f(x) = \frac{x^2$ - 4}{x - 2}$$

Question 19

Given $$f(x) = \frac{1}{x}$$, which of the following statements is true about its symmetry?

Accepted Answer

It is an odd function, symmetric about the origin.

Question 20

If $$f(x) = 3x + 2$$ and $$g(x) = x^2$, what is the domain of f$$ \circ g$$?

Accepted Answer

$$(-\infty, \infty)$$

Precalculus Midterm Study Notes: Trigonometry, Functions, Logarithms, and More

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