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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