Q: Given the rational function $f(x) = \frac{x^2 + 4x + 3}{x^2 - 9}$, what is its domain?

$\{x \mid x \neq 3, x \neq -3\}$

Q: Given $f(x) = \frac{3x}{x^2 + 12}$, what is the domain?

$\{x \mid x \neq \pm 2\sqrt{3}\}$

Q: Simplify $f(x) = \frac{x^2 - 9}{x - 3}$ and state the domain.

$f(x) = x + 3$, domain: $x \neq 3$

Question 1

Which of the following best describes a rational function?

Accepted Answer

A function with a polynomial in the numerator and a polynomial in the denominator.

Question 2

Given the rational function $f(x) = \frac{x^2 + 4x + 3}{x^2 - 9}$, what is its domain?

Accepted Answer

$\{x \mid x 
eq 3, x 
eq -3\}$

Question 3

What is the lowest terms form of $f(x) = \frac{x^2 + 5x + 6}{x^2 - 25}$?

Accepted Answer

$\frac{(x+2)(x+3)}{(x+5)(x-5)}$

Question 4

For the rational function $f(x) = \frac{6x}{2x^2 - 8}$, which value(s) must be excluded from the domain?

Accepted Answer

$x = 2, x = -2$

Question 5

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

Accepted Answer

$\{x \mid x 
eq 3\}$

Question 6

Which step is necessary before writing a rational function in lowest terms?

Accepted Answer

Find the domain by setting the denominator not equal to zero.

Question 7

Given $f(x) = \frac{3x}{x^2 + 12}$, what is the domain?

Accepted Answer

$\{x \mid x 
eq \pm 2\sqrt{3}\}$

Question 8

A rational function $f(x) = \frac{p(x)}{q(x)}$ is undefined when:

Accepted Answer

The denominator $q(x)$ equals zero.

Question 9

Simplify $f(x) = \frac{x^2 - 9}{x - 3}$ and state the domain.

Accepted Answer

$f(x) = x + 3$, domain: $x 
eq 3$

Question 10

Which of the following is NOT a rational function?

Accepted Answer

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

Question 11

Given $f(x) = \frac{x^2 + 4x + 3}{x^2 - 9}$, which values of $x$ make the denominator zero?

Accepted Answer

$x = 3$ and $x = -3$

Question 12

A medical dosage formula is given by $D(x) = \frac{2x + 5}{x - 1}$, where $x$ is the patient's weight in kg. For which weight is the formula undefined?

Accepted Answer

$x = 1$

Question 13

If $f(x) = \frac{x^2 - 4}{x^2 - 8x + 16}$, what is the domain?

Accepted Answer

$\{x \mid x 
eq 4\}$

Question 14

A rational function $f(x)$ is written in lowest terms when:

Accepted Answer

All common factors in the numerator and denominator are cancelled.

Question 15

Given $f(x) = \frac{2x^2 + 4x + 2}{x^2 - 1}$, what is the lowest terms form?

Accepted Answer

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

Question 16

A chemical reaction rate is modeled by $R(x) = \frac{x^2 - 16}{x^2 - 4x}$. For which value(s) of $x$ is the rate undefined?

Accepted Answer

$x = 0$ and $x = 4$

Question 17

If $f(x) = \frac{x^2 + 9}{x - 3}$, what is the lowest terms form?

Accepted Answer

$\frac{x^2 + 9}{x - 3}$

Question 18

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 exclude those $x$ values from the domain.

Rational Functions: Definitions, Domains, and Simplification

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