Q: Which of the following best describes the domain of the function $f(x) = \frac{x}{2x^2 + x}$?

$x \neq 0,\ x \neq -\frac{1}{2}$

Question 1

Which of the following best describes the domain of the function $f(x) = \frac{x}{2x^2 + x}$?

Accepted Answer

$x 
eq 0,\ x 
eq -\frac{1}{2}$

Question 2

Simplify the rational expression $\frac{x^2 - 81}{9 + x}$.

Question 3

A publishing company models the cost per book for printing $x$ books as $C(x) = \frac{500 + 2x}{x}$. What is the cost per book for printing 1000 books?

Accepted Answer

$2.50$

Question 4

Which of the following is the simplified form of $\frac{y^2 + 5y + 6}{y^2 + 10y + 21}$?

Accepted Answer

$\frac{y + 3}{y + 7}$

Question 5

Multiply and simplify: $\frac{24xy^2}{x^2 - 49} \cdot \frac{3x - 21}{8x^2y^2}$.

Accepted Answer

$\frac{9}{8(x + 7)}$

Question 6

Find the least common denominator (LCD) for the rational expressions $\frac{6}{7y}$ and $\frac{3}{y^2}$.

Accepted Answer

$7y^2$

Question 7

Add and simplify: $\frac{7}{x^2} + \frac{3}{x}$.

Accepted Answer

$\frac{7 + 3x}{x^2}$

Question 8

Subtract and simplify: $\frac{x^2}{x + 3} - \frac{9}{x + 3}$.

Accepted Answer

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

Question 9

Which of the following is the correct LCD for $\frac{6}{x^2 - y^2}$ and $\frac{1}{x^2 + 2xy + y^2}$?

Accepted Answer

$(x - y)(x + y)^2$

Question 10

A medical laboratory uses the formula $C(x) = \frac{1000 + 5x}{x}$ to determine the cost per test for $x$ tests. If 200 tests are performed, what is the cost per test?

Accepted Answer

$10$

Question 11

Simplify the expression $\frac{8}{x - 1} - \frac{14}{x^2 - 1}$.

Question 12

Which of the following is the simplified form of $\frac{6x + x^2}{5x}$?

Accepted Answer

$\frac{x + 6}{5}$

Question 13

Add and simplify: $\frac{x + 3}{x^2 + 4x - 12} + \frac{x - 2}{x^2 - 4}$.

Accepted Answer

$\frac{2x + 1}{(x + 6)(x - 2)}$

Question 14

Subtract and simplify: $\frac{1}{-x} - \frac{1}{x}$.

Accepted Answer

$-\frac{2}{x}$

Question 15

Which of the following is the correct LCD for $\frac{1}{x^2}$ and $\frac{1}{x}$?

Accepted Answer

$x^2$

Question 16

A chemical reaction's rate is modeled by $R(t) = \frac{500}{t + 10}$. What is the rate when $t = 40$ seconds?

Accepted Answer

$12.5$

Question 17

Which of the following is the simplified form of $\frac{x^2 + 5x - 6}{x^2 + 9x + 18} + \frac{x^2 - 1}{x^2 + 7x + 12}$?

Accepted Answer

$\frac{2x + 1}{(x + 6)(x - 2)}$

Question 18

If $g(x) = -\frac{6x + x^2}{5x}$, what is the value of $g(-5)$?

Accepted Answer

$2$

Question 19

A company’s average cost per item is given by $A(x) = \frac{1200 + 8x}{x}$. If the company wants the average cost per item to be $20$, how many items must be produced?

Accepted Answer

$60$

Rational Functions: Multiplication, Division, Addition, and Subtraction of Rational Expressions

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