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: 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

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

E1

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