$$\frac{x-2}{x}$$, $$x \neq 0, x \neq 2$$

$$\frac{x+5}{x-5}$$, $$x \neq 5$$

$$\frac{x-2}{x+2}$$, $$x \neq 2, x \neq -2$$

Question 1

Given the rational function $$f(x) = \frac{x^2$ - 9}{x^2$ - 4}$$, which of the following describes its domain?

Accepted Answer

All real numbers except $$x = 2$$ and $$x = -2$$

Question 2

For the function $$f(x) = \frac{(x-3)(x+2)}{(x-3)(x-1)}$$, what type of discontinuity occurs at $$x = 3$$?

Accepted Answer

A hole at $$x = 3$$

Question 3

Simplify $$\frac{x^2$ - 4x + 4}{x^2$ - 2x}$$ and state any restrictions on the domain.

Accepted Answer

$$\frac{x-2}{x}$$, $$x 
eq 0, x 
eq 2$$

Question 4

Which of the following rational functions has a vertical asymptote at $$x = -5$$?

Accepted Answer

$$f(x) = \frac{2x}{x+5}$$

Question 5

If $$f(x) = \frac{x^2$ - 1}{x^2$ - 4x + 3}$$, what are the vertical asymptotes?

Accepted Answer

$$x = 1$$ and $$x = 3$$

Question 6

A rational function $$f(x)$$ has a hole at $$x = 2$$ and a vertical asymptote at $$x = -3. $$Which of the following could be $$f(x)$$?

Accepted Answer

$$\frac{(x-2)(x+1)}{(x-2)(x+3)}$$

Question 7

Divide $$\frac{x^2$ + 5x + 6}{x + 2}$$ and express the result as a polynomial plus a remainder.

Accepted Answer

$$x + 3 + \frac{0}{x+2}$$

Question 8

Which operation is required to combine $$\frac{2}{x-3}$$ and $$\frac{5}{x+2}$$ into a single rational expression?

Accepted Answer

Find a common denominator and add numerators

Question 9

A medical dosage formula is $$D(x) = \frac{100}{x-4}$$, where $$x is $$the patient's weight in kg. For which value of $$x is $$the formula undefined?

Accepted Answer

$$x = 4$$

Question 10

Simplify $$\frac{x^2$ - 25}{x^2$ - 10x + 25}$$ and state the domain.

Accepted Answer

$$\frac{x+5}{x-5}$$, $$x 
eq 5$$

Question 11

If $$f(x) = \frac{x^2$ - 4x + 3}{x^2$ - 9}$$, what is the value of $$f(3)$$?

Accepted Answer

Undefined (hole or vertical asymptote)

Question 12

A rational function models the concentration of a chemical: $$C(t) = \frac{5t}{t^2$ - 16}. $$What is the vertical asymptote?

Accepted Answer

$$t = 4$$ and $$t = -4$$

Question 13

Which of the following is the result of multiplying $$\frac{x+1}{x-2} by$$ $$\frac{x-2}{x+3}$$?

Accepted Answer

$$\frac{x+1}{x+3}$$

Question 14

A rational function $$f(x) = \frac{x^2$ - 6x + 9}{x^2$ - 9} is $$used to model a population. What is the domain?

Accepted Answer

All real numbers except $$x = 3$$ and $$x = -3$$

Question 15

If $$f(x) = \frac{x^2$ - 4}{x^2$ - 4x + 4}$$, what type of discontinuity occurs at $$x = 2$$?

Accepted Answer

A hole at $$x = 2$$

Question 16

Divide $$\frac{x^3$ - 8}{x - 2}$$ using long division. What is the quotient?

Accepted Answer

$$x^2 + 2x + 4$

Question 17

Which of the following is the result of adding $$\frac{3}{x}$$ and $$\frac{2}{x+1}$$?

Accepted Answer

$$\frac{3(x+1) + 2x}{x(x+1)}$$

Question 18

A rational function $$f(x) = \frac{x^2$ - 1}{x^2$ - 2x + 1} is $$used in a physics experiment. What is the domain?

Accepted Answer

All real numbers except $$x = 1$$

Question 19

If $$f(x) = \frac{x^2$ - 4x + 4}{x^2$ - 4}$$, what is the simplified form and domain?

Accepted Answer

$$\frac{x-2}{x+2}$$, $$x 
eq 2, x 
eq -2$$

Rational Functions: Properties, Operations, and Applications

Study Guide - Practice Questions