$$\frac{\sqrt{x+4}}{x-1}$$, \quad x \geq -4, \ x \neq 1$

Question 1

Given the set of ordered pairs $$\{(-3, 2), (2, 1), (4, 3), (1, -3)\}$$, what is the domain of the relation?

Accepted Answer

$$\{-3, 2, 4, 1\}$$

Question 2

Which of the following best describes the range of the relation $$\{(-3, 2), (2, 1), (4, 3), (1, -3)\}$$?

Accepted Answer

$$\{2, 1, 3, -3\}$$

Question 3

Which of the following relations is a function?

Accepted Answer

$$\{(2, 5), (-3, 7), (4, 5), (0, -1)\}$$

Question 4

Given the graph of $$f(x) = |x|$$, which transformation results in the graph of $$g(x) = |x| + 3$$?

Accepted Answer

A vertical shift up by 3 units

Question 5

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

Accepted Answer

$$y = (2x + 5)^2$

Question 6

Given $$g(x) = \sqrt{x-5}$$, what is the domain of $$g(x)$$?

Accepted Answer

$$x \geq 5$$

Question 7

Which of the following graphs passes the vertical line test and represents a function?

Accepted Answer

A parabola opening upwards

Question 8

If $$f(x) = x^2$ + 6$$, determine if $$f(x) is $$even, odd, or neither.

Accepted Answer

Even

Question 9

Given $$f(x) = x + 2$$ and $$g(x) = 3x + 5$$, find $$(f \cdot g)(x).$$

Accepted Answer

$$(x + 2)(3x + 5)$$

Question 10

Given $$f(x) = \sqrt{x+4}$$ and $$g(x) = x-1$$, find $$\left(\frac{f}{g}\right)(x)$$ and its domain.

Accepted Answer

$$\frac{\sqrt{x+4}}{x-1}$$, \quad x \geq -4, \ x 
eq 1$

Question 11

Which of the following is the correct difference quotient for $$f(x) = 2 + 4x$$?

Accepted Answer

$$\frac{f(x+h) - f(x)}{h} = 4$$

Question 12

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

Accepted Answer

$$x \geq 5$$

Question 13

Which of the following describes a function that is symmetric with respect to the y-axis?

Accepted Answer

Even function

Question 14

If $$f(x) = x^2$ + 1$$ for $$x \geq 0$$, what is the inverse function $$f^{-1}(x$$)$$?

Accepted Answer

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

Question 15

Which of the following is NOT a valid reason for restricting the domain of a function?

Accepted Answer

The function is defined for all real numbers

Question 16

Given the piecewise function $$f(x) = \begin{cases} -2x + 4 & x \leq 1 \ 3 & 1 \u003c x \u003c 4 \ x + 1 & x \geq 4 \end{cases}$$, what is $$f(5)$$?

Accepted Answer

$$6$$

Question 17

Which of the following is the correct order for performing multiple transformations on a function?

Accepted Answer

1. Horizontal shifts, 2. Stretch/Shrink, 3. Reflections, 4. Vertical shifts

Question 18

If $$f(x) = 4x - 7$$ and $$g(x) = \frac{x+7}{4}$$, what is $$(f \circ g)(x)$$?

Accepted Answer

$$x$$

Question 19

Which of the following graphs is symmetric with respect to the origin?

Accepted Answer

$$y = x^3$

Comprehensive Study Notes: Introduction to Functions and Their Properties

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