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 $$\frac{x^2$ - 81}{9 + x}.$$

Accepted Answer

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

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

Accepted Answer

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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