$$f(x) = \begin{cases} x, & x \u003c 1 \\ -2x+5, & 1 \leq x \u003c 2 \\ -2, & x \geq 2 \end{cases}$$

Question 1

Given the piecewise function represented by the graph below, which of the following is the correct algebraic rule for $$f(x)$$?

Accepted Answer

$$f(x) = \begin{cases} x, & x \u003c 1 \ -2x+5, & 1 \leq x \u003c 2 \ -2, & x \geq 2 \end{cases}$$

Question 2

Which of the following best describes the amplitude and period of the trigonometric function shown in the second image?

Accepted Answer

Amplitude: $$4$$, Period: $$\pi$$

Question 3

Given the tangent function graph in the third image, what is the period of the function?

Accepted Answer

$$2\pi$$

Question 4

Which transformation will produce the graph of $$k(x) = 5f(-x+3) + 1$$ from $$f(x)$$?

Accepted Answer

Reflect about the y-axis, shift right 3 units, vertical stretch by 5, shift up 1

Question 5

If $$	an 	heta \u003e 0$$ and $$\sin 	heta \u003c 0$$, in which quadrant does $$	heta$$ lie?

Accepted Answer

Quadrant III

Question 6

Simplify the logarithmic expression: $$12\ln y - 4\ln w + 2\ln x - \ln z$$

Accepted Answer

$$\ln \left( \dfrac{y$$^{12}$$x^2$$}{$$w^4z$$} \right)$

Question 7

Given $$\csc(\beta) = -3$$ and $$	an(\beta) \u003c 0$$, what is $$\cos(\beta)$$?

Accepted Answer

$$\dfrac{2}{3}$$

Question 8

Find the average rate of change of $$f(x) = -3x^2$ + 2x$$ from $$x = 1 to$$ $$x = 3.$$

Accepted Answer

$$-14$$

Question 9

Which of the following is the correct domain for $$f(x) = \ln(x+6) + 4 - x$$?

Accepted Answer

$$x \u003e -6$$

Question 10

Given $$f(x) = 4x - 2x + 5$$, what is the end behavior as $$x 	o \infty$$?

Accepted Answer

$$f(x) 	o \infty$$

Question 11

Find the vertex of $$p(t) = -3t^2$ + 6t + 12.$$

Accepted Answer

$$(1, 15)$$

Question 12

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

Accepted Answer

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

Question 13

If $$\log_a 2 = 0.4$$, $$\log_a 3 = 0.5$$, $$\log_a 5 = 0.7$$, and $$\log_a 7 = 0.8$$, what is $$\log_a \left( \dfrac{21}{210} \right)$$?

Accepted Answer

$$-1.1$$

Question 14

A rectangle has area $$32\ \mathrm{ft}^2$$ and its length is twice its width. What is the length of the diagonal?

Accepted Answer

$$8\sqrt{2} ft$$

Question 15

Which of the following is the correct simplification for $$\ln \left( \dfrac{y^4$}{3x^5$} \right)$$?

Accepted Answer

$$4\ln y - \ln 3 - 5\ln x$$

Question 16

Given $$f(x) = \log_2$ x$$, which transformation produces $$k(x) = 5\log_2$(x+1) - 3$$?

Accepted Answer

Vertical stretch by 5, shift left 1, shift down 3

Question 17

Solve for $$x$$: $$\ln(x+3) + \ln(x-3) = \ln 7$$

Accepted Answer

$$x = 4$$

Question 18

Given $$f(x) = x^2$ + 6x + 4 - x$$, what is the domain of $$f(x)$$?

Accepted Answer

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

Question 19

Which of the following is the correct factorization for $$g(x) = 3x^4$ - 14x^3$ - 5x^2$?

Accepted Answer

$$x^2(3x^2$ - 14x - 5)$

Precalculus Review: Graph Interpretation and Function Analysis

Study Guide - Practice Questions