$$(f \circ g)(x) = (\sqrt{x-8})^2$ = x-8$$, \text{ domain: } [8, \infty)$

Question 1

Which of the following best describes a function?

Accepted Answer

A relation where each input has at most one output.

Question 2

Given the relation $$\{(-3, 5), (0, 2), (3, 5)\}$$, what are the inputs and outputs, and is this a function?

Accepted Answer

Inputs: $$\{-3, 0, 3\}$$, Outputs: $$\{2, 5\}$$, Yes, it is a function.

Question 3

Which equation represents a function?

Accepted Answer

$$y + 4 = 3x$$

Question 4

What is the domain of the function $$f(x) = \sqrt{x+4}$$? Express your answer in interval notation.

Accepted Answer

$$[-4, \infty)$$

Question 5

Given $$f(x) = \frac{1}{x^2$ - 5x + 6}$$, what is the domain of $$f(x) in $$interval notation?

Accepted Answer

$$(-\infty, 2) \cup (2, 3) \cup (3, \infty)$$

Question 6

Which transformation is represented by $$f(x) \rightarrow f(x-3) + 2$$?

Accepted Answer

A shift right by 3 units and up by 2 units.

Question 7

If $$f(x) = x^2$, what is the equation for the function reflected over the x-axis and shifted down 2 units?

Accepted Answer

$$g(x) = -x^2$ - 2$$

Question 8

Given the line passing through the points $$(-1, 1)$$ and $$(4, 3)$$, what is its slope?

Accepted Answer

$$m = \frac{3 - 1}{4 - (-1)} = \frac{2}{5}$$

Question 9

Which of the following is the equation of a circle with center $$(1, -2)$$ and radius $$3$$?

Accepted Answer

$$(x-1)^2$$ + $$(y+2)^2 = 9$

Question 10

Given $$f(x) = x^2$ and g(x$$) = \sqrt{x-8}$$, find (f$$ \circ g)(x)$$ and its domain.

Accepted Answer

$$(f \circ g)(x) = (\sqrt{x-8})^2$ = x-8$$, 	ext{ domain: } [8, \infty)$

Question 11

Which of the following is NOT a function?

Accepted Answer

$$x^2$$ + $$y^2 = 9$

Question 12

Given $$f(x) = x^2$ + \frac{1}{x}$$ and $$g(x) = x^2$ + x + 2$$, what is the domain of $$f(x) + g(x)$$?

Accepted Answer

$$(-\infty, 0) \cup (0, \infty)$$

Question 13

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

Accepted Answer

$$f(g(2)) = f(4) = 4 + 3 = 7$$

Question 14

Which transformation results in a vertical stretch of a function $$f(x) by a $$factor of 2?

Accepted Answer

$$2 \cdot f(x)$$

Question 15

Given the equation $$x^2$$ + $$y^2 + 2x - 4y + 1 = 0$, what is the center and radius of the circle after converting to standard form?

Accepted Answer

Center: $$(-1, 2)$$, Radius: $$2$$

Question 16

Which of the following best describes the range of the function $$f(x) = x^2$?

Accepted Answer

$$[0, \infty)$$

Question 17

If $$f(x) = x^2$ and g(x) = x + 1$, what is (g$$ \circ f)(2)$$?

Accepted Answer

$$g(f(2)) = g(4) = 4 + 1 = 5$$

College Algebra Essentials: Functions, Graphs, and Transformations

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