Q: Simplify the following rational expression and state any restrictions on the domain: $$\frac{x^2$ - 4}{x^2$ - x - 6}$$

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

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

$$\frac{x+3}{x-3}$$, $$x \neq 3$$

Q: If $$f(x) = \frac{x^2$-1}{x^2$+2x+1}$$, what is the simplified form of $$f(x)$$ and its domain restriction?

$$\frac{x-1}{x+1}$$, $$x \neq -1$$

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

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

Question 1

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

Accepted Answer

A function of the form $$f(x) = \frac{p(x)}{q(x)}$$ where $$p(x)$$ and $$q(x)$$ are polynomials and $$q(x) 
eq 0.$$

Question 2

Simplify the following rational expression and state any restrictions on the domain: $$\frac{x^2$ - 4}{x^2$ - x - 6}$$

Accepted Answer

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

Question 3

What is the least common denominator (LCD) for the rational expressions $$\frac{1}{x}$$ and $$\frac{2}{x+4}$$?

Accepted Answer

$$x(x+4)$$

Question 4

Which of the following is the correct result of multiplying $$\frac{2x}{x^2$-9} by$$ $$\frac{x+3}{2x}$$?

Accepted Answer

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

Question 5

Divide $$\frac{x^2$-1}{x+2} by$$ $$\frac{x-1}{x+2}$$ and simplify your answer.

Accepted Answer

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

Question 6

Which step is necessary when adding $$\frac{2}{x}$$ and $$\frac{3}{x+1}$$?

Accepted Answer

Find the least common denominator (LCD) of $$x$$ and $$x+1.$$

Question 7

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

Accepted Answer

$$x = -2$$

Question 8

Which of the following is a restriction on the domain of $$f(x) = \frac{x+2}{x^2$-4}$$?

Accepted Answer

$$x 
eq 2$$ and $$x 
eq -2$$

Question 9

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

Accepted Answer

$$\frac{x+3}{x-3}$$, $$x 
eq 3$$

Question 10

Given $$f(x) = \frac{x^2$-4}{x^2$+2x}$$, which values of $$x$$ must be excluded from the domain?

Accepted Answer

$$x 
eq 0, x 
eq -2$$

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

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

Accepted Answer

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

Question 13

If $$f(x) = \frac{x^2$-1}{x^2$+2x+1}$$, what is the simplified form of $$f(x)$$ and its domain restriction?

Accepted Answer

$$\frac{x-1}{x+1}$$, $$x 
eq -1$$

Question 14

Solve for $$x$$: $$\frac{3}{x-2} = \frac{5}{x+3}$$

Accepted Answer

$$x = 7$$

Question 15

Which of the following is a correct step when solving a rational equation?

Accepted Answer

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

Question 16

A chemical reaction rate is modeled by $$R(t) = \frac{t^2$-4}{t-2}. $$What is the simplified form of $$R(t)$$ and its domain restriction?

Accepted Answer

$$t+2$$, $$t 
eq 2$$

Question 17

Which of the following is NOT a valid step when simplifying $$\frac{x^2$-9}{x^2$-6x+9}$$?

Accepted Answer

Cancel $$x$$ terms directly without factoring.

Question 18

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

Accepted Answer

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

Question 19

Which of the following is the correct result of subtracting $$\frac{2}{x}$$ from $$\frac{5}{x+1}$$?

Accepted Answer

$$\frac{5x-2(x+1)}{x(x+1)}$$

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