Q: Simplify the expression $$\frac{x^2$ - 9}{x^2$ - 6x + 9}$$ and state any restrictions on the domain.

$$\frac{(x+3)(x-3)}{(x-3)^2$}$$, $$x \neq 3$$

Q: When dividing $$\frac{x^2$ + x}{x^2$ - 1} by$$ $$\frac{x + 1}{x - 1}$$, what is the simplified result?

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

Q: Simplify $$\frac{2x}{x^2$ - 4} + \frac{3}{x - 2}$$ and state any restrictions on the domain.

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

Q: Simplify $$\frac{x^2$ - 4x}{x^2$ - 16}$$ and state any restrictions on the domain.

$$\frac{x(x - 4)}{(x - 4)(x + 4)}$$, $$x \neq 4, -4$$

Question 1

Which of the following is the correct definition of a rational function?

Accepted Answer

A function that can be written as $$f(x) = \frac{p(x)}{q(x)}$$ where $$p(x)$$ and $$q(x)$$ are polynomials and $$q(x) 
eq 0.$$

Question 2

What is the domain of the rational function $$f(x) = \frac{x^2$ - 4}{x^2$ - x - 6}$$?

Accepted Answer

All real numbers except $$x = 2$$ and $$x = -3.$$

Question 3

Simplify the expression $$\frac{x^2$ - 9}{x^2$ - 6x + 9}$$ and state any restrictions on the domain.

Accepted Answer

$$\frac{(x+3)(x-3)}{(x-3)^2$}$$, $$x 
eq 3$$

Question 4

Which of the following is the correct result of multiplying $$\frac{2x}{x^2$ - 4}$$ and $$\frac{x^2$ - 4}{x + 2}$$?

Accepted Answer

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

Question 5

When dividing $$\frac{x^2$ + x}{x^2$ - 1} by$$ $$\frac{x + 1}{x - 1}$$, what is the simplified result?

Accepted Answer

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

Question 6

What is the least common denominator (LCD) for the rational expressions $$\frac{1}{x}$$ and $$\frac{2}{x + 1}$$?

Accepted Answer

$$x(x + 1)$$

Question 7

Simplify $$\frac{2x}{x^2$ - 4} + \frac{3}{x - 2}$$ and state any restrictions on the domain.

Accepted Answer

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

Question 8

Solve for $$x$$: $$\frac{2}{x} = \frac{3}{x + 1}.$$

Accepted Answer

$$x = -2$$

Question 9

Which of the following is a necessary step when solving rational equations?

Accepted Answer

Multiply both sides by the least common denominator (LCD).

Question 10

Given $$f(x) = \frac{x^2$ - 1}{x^2$ + x}$$, which values must be excluded from the domain?

Accepted Answer

$$x = 0$$ and $$x = -1$$

Question 11

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

Accepted Answer

$$x = 5$$

Question 12

Simplify $$\frac{x^2$ - 4x}{x^2$ - 16}$$ and state any restrictions on the domain.

Accepted Answer

$$\frac{x(x - 4)}{(x - 4)(x + 4)}$$, $$x 
eq 4, -4$$

Question 13

A chemical reaction rate is modeled by $$R(t) = \frac{t^2$ - 9}{t - 3}. $$What is the simplified form of $$R(t)$$ for $$t 
eq 3$$?

Accepted Answer

$$t + 3$$

Question 14

Which of the following is NOT a valid step when adding $$\frac{2}{x}$$ and $$\frac{3}{x + 1}$$?

Accepted Answer

Add the numerators directly: $$2 + 3$$

Question 15

Solve for $$x$$: $$\frac{x}{x - 2} = 3.$$

Accepted Answer

$$x = 6$$

Question 16

Which of the following expressions cannot be reduced further?

Accepted Answer

$$\frac{x^2$ + x + 1}{x + 1}$$

Question 17

A rational function $$f(x)$$ has a denominator $$x^2 - 9$. Which values of x$ must be excluded from the domain?

Accepted Answer

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

Question 18

If $$\frac{a}{b} + \frac{c}{d} is to be $$written as a single fraction, what is the correct formula?

Accepted Answer

Question 19

Which of the following is the correct result of subtracting $$\frac{3}{x}$$ from $$\frac{5}{x}$$?

Accepted Answer

$$\frac{2}{x}$$

E1

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