Question 1

Given the relation $$\{(2,3), (4,5), (8,8)\}$$, determine whether it is a function. What are the domain and range?

Accepted Answer

It is a function. Domain: $$\{2, 4, 8\}$$, Range: $$\{3, 5, 8\}$$

Question 2

For the function $$f(x) = x^2$ - 13x + 4$$, what is the domain?

Accepted Answer

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

Question 3

Find the domain of $$f(x) = \sqrt{x+5}.$$

Accepted Answer

$$[-5, \infty)$$

Question 4

Find the domain of $$f(x) = \frac{1}{\sqrt{x+11}}.$$

Accepted Answer

$$(-11, \infty)$$

Question 5

Given $$g(x) = \frac{\sqrt{x-2}}{x-9}$$, what is the domain of $$g$$?

Accepted Answer

$$[2, 9) \cup (9, \infty)$$

Question 6

Given $$f(x) = \frac{3x+10}{x^2$ + 2x^2$ - 9x - 18}$$, find the domain of $$f.$$

Accepted Answer

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

Question 7

Suppose $$f(x) = x^2$ - 5x + 5. $$Find and simplify the difference quotient $$\frac{f(x+h) - f(x)}{h}.$$

Accepted Answer

$$2x + h - 5$$

Question 8

Given $$f(x) = 6x$$ and $$g(x) = x + 7$$, find $$(f \circ g)(x).$$

Accepted Answer

$$6x + 42$$

Question 9

Given $$f(x) = 3x + 3$$ and $$g(x) = \frac{x-3}{3}$$, are $$f$$ and $$g$$ inverses of each other?

Accepted Answer

Yes, $$f$$ and $$g$$ are inverses of each other.

Question 10

Which of the following equations defines $$y as a $$function of $$x$$?

Accepted Answer

$$x + y = 41$$

Question 11

Given the function $$f(x) = \sqrt{x-3} - 4$$, what is the domain of $$f$$?

Accepted Answer

$$[3, \infty)$$

Question 12

Given the relation $$\{(4,1), (4,5), (6,1), (6,5)\}$$, is this a function? What is the domain?

Accepted Answer

No, domain is $$\{4, 6\}$$

Question 13

If $$f(x) = x^2$ + 6x + 7$$, what is $$f(x+7)$$?

Accepted Answer

$$x^2 + 20x + 98$

Question 14

Which of the following graphs passes the vertical line test and thus represents $$y as a $$function of $$x$$?

Accepted Answer

A graph where every vertical line intersects the curve at most once.

Question 15

Given $$f(x) = x^2$ - 13x + 4$$, what is the range of $$f$$?

Accepted Answer

$$[k, \infty)$$ where $$k is $$the minimum value of $$f$$

Question 16

Given $$f(x) = x^2$ - 5x + 5$$ and $$g(x) = 3x^2$, what is the domain of $$\frac{f}{g}$$?

Accepted Answer

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

Question 17

Given $$f(x) = x^2$ - 13x + 4$$, what is the $$x-$$value of the vertex?

Accepted Answer

$$x = \frac{13}{2}$$

Question 18

Given $$f(x) = x^2$ - 13x + 4$$, what is the $$y-$$value of the vertex?

Accepted Answer

$$y = 4 - 13 \cdot \frac{13}{2} + \left(\frac{13}{2}\$$right)^2$$

Question 19

Given $$f(x) = x^2$ - 13x + 4$$, for which values of $$x is$$ $$f(x)$$ increasing?

Accepted Answer

$$\left(\frac{13}{2}, \infty\right)$$

College Algebra Study Guide: Functions, Domains, Ranges, and Transformations

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