Q: Simplify $$f(x) = \frac{x^2$-1}{x-1}$$ and state its domain.

$$f(x) = x+1$$, domain: $$x \neq 1$$

Q: Write $$f(x) = \frac{x^2$-4}{x^2$+2x-8} in $$lowest terms and state its domain.

$$f(x) = \frac{x+2}{x+4}$$, domain: $$x \neq 2, x \neq -4$$

Q: Given $$f(x) = \frac{2x^2$-8}{x^2$-4}$$, write $$f(x) in $$lowest terms and state the domain.

$$f(x) = 2$$, domain: $$x \neq 2, x \neq -2$$

Q: If $$f(x) = \frac{5x-10}{x^2$-4}$$, what is $$f(x) in $$lowest terms?

$$\frac{5}{x+2}$$, \quad x \neq 2, x \neq -2$

Question 1

Which of the following best describes a rational function?

Accepted Answer

A function with a polynomial in the numerator and denominator.

Question 2

Given $$f(x) = \frac{12}{x-1}$$, what is the domain of $$f(x)$$?

Accepted Answer

$$x 
eq 1$$

Question 3

Which value of $$x$$ makes the denominator of $$f(x) = \frac{1}{x^2$-1}$$ equal to zero?

Accepted Answer

$$x = 1 or$$ $$x = -1$$

Question 4

Simplify $$f(x) = \frac{x^2$-1}{x-1}$$ and state its domain.

Accepted Answer

$$f(x) = x+1$$, domain: $$x 
eq 1$$

Question 5

Which of the following is NOT a rational function?

Accepted Answer

$$f(x) = \sqrt{x}$$

Question 6

Find the domain of $$f(x) = \frac{2x}{x^2$-4}.$$

Accepted Answer

$$x 
eq 2, x 
eq -2$$

Question 7

Write $$f(x) = \frac{x^2$-4}{x^2$+2x-8} in $$lowest terms and state its domain.

Accepted Answer

$$f(x) = \frac{x+2}{x+4}$$, domain: $$x 
eq 2, x 
eq -4$$

Question 8

If $$f(x) = \frac{3x+2}{x-5}$$, which value is excluded from the domain?

Accepted Answer

$$x = 5$$

Question 9

A rational function models the concentration $$C(t) = \frac{100}{t-2} of a $$chemical in a solution, where $$t is $$time in hours. When is the concentration undefined?

Accepted Answer

At $$t = 2$$ hours

Question 10

Which of the following rational functions has a domain of all real numbers except $$x = -3$$ and $$x = 2$$?

Accepted Answer

$$f(x) = \frac{x+1}{x^2$+x-6}$$

Question 11

Given $$f(x) = \frac{2x^2$-8}{x^2$-4}$$, write $$f(x) in $$lowest terms and state the domain.

Accepted Answer

$$f(x) = 2$$, domain: $$x 
eq 2, x 
eq -2$$

Question 12

Which step is necessary when simplifying a rational function?

Accepted Answer

Factor numerator and denominator, then cancel common factors.

Question 13

If $$f(x) = \frac{5x-10}{x^2$-4}$$, what is $$f(x) in $$lowest terms?

Accepted Answer

$$\frac{5}{x+2}$$, \quad x 
eq 2, x 
eq -2$

Question 14

A rational function $$f(x)$$ has a denominator $$x^2+2x-8$. Which values must be excluded from the domain?

Accepted Answer

$$x = -4, x = 2$$

Question 15

Which of the following is the correct process to find the domain of a rational function?

Accepted Answer

Set the denominator equal to zero and solve for $$x.$$

Introduction to Rational Functions: Definitions, Domains, and Simplification

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