Question 1

Which of the following is the correct domain for the rational function $$f(x) = \dfrac{3x}{x-7}$$?

Accepted Answer

$$x 
eq 7$$

Question 2

Simplify the rational expression $$\dfrac{x^2$ - 8x}{x^2$ - 16}.$$

Accepted Answer

$$\dfrac{x(x-8)}{(x-4)(x+4)}$$

Question 3

Multiply and simplify: $$\dfrac{x+3}{x^2$-9} \cdot \dfrac{x-3}{x+3}.$$

Accepted Answer

$$\dfrac{1}{x-3}$$

Question 4

Divide and simplify: $$\dfrac{x^2$-9}{x+2} \div \dfrac{x-3}{x-2}.$$

Accepted Answer

$$\dfrac{(x+3)(x-2)}{x+2}$$

Question 5

A book's total revenue is modeled by $$R(x) = \dfrac{1000x^2$}{x^2$+4}$$, where $$x is $$years since publication. What is the total revenue at the end of the first year (in millions)?

Accepted Answer

$$250$$

Question 6

Add the rational expressions: $$\dfrac{4}{x+3} + \dfrac{3}{x+3}.$$

Accepted Answer

$$\dfrac{7}{x+3}$$

Question 7

Find the least common denominator (LCD) for the expressions $$\dfrac{1}{a^2$b}$$ and $$\dfrac{2}{ab^2$}.$$

Accepted Answer

$$a^2b^2$

Question 8

Simplify the complex fraction: $$\dfrac{\dfrac{2}{x} + \dfrac{1}{y}}{\dfrac{1}{x} - \dfrac{1}{y}}.$$

Accepted Answer

$$\dfrac{2y + x}{x - y}$$

Question 9

Divide using synthetic division: $$2x^3$$ - $$7x^2 + 13x - 5$ by x-2$. What is the remainder?

Accepted Answer

$$3$$

Question 10

Solve for $$x$$: $$\dfrac{2}{x} + 1 = 5.$$

Accepted Answer

Question 11

The sum of a number and five times its reciprocal is 6. Which equation models this situation?

Accepted Answer

$$x + \dfrac{5}{x} = 6$$

Question 12

An Arabian camel can drink 15 gallons of water in 10 minutes. At this rate, how much water can it drink in 3 minutes?

Accepted Answer

$$4.5$$ gallons

Question 13

Three postal workers can sort a stack of mail in 20, 30, and 60 minutes, respectively. How long will it take them to sort the mail together?

Accepted Answer

$$12$$ minutes

Question 14

If the speed of a river's current is 5 mph, and a boat travels 20 miles downstream in the same time it takes to travel 10 miles upstream, what is the speed of the boat in still water?

Accepted Answer

$$15$$ mph

Question 15

If $$y$$ varies directly as $$x$$ and $$y = 4$$ when $$x = 20$$, what is the constant of variation $$k$$?

Accepted Answer

$$k = 0.2$$

Question 16

If $$y$$ varies inversely as $$x$$ and $$y = 6$$ when $$x = 5$$, what is the inverse variation equation?

Accepted Answer

$$y = \dfrac{30}{x}$$

Question 17

The weight $$W of a $$synthetic ball varies directly with the cube of its radius $$r. If a $$ball with radius $$2$$ inches weighs $$1.20$$ pounds, what is the weight of a ball with radius $$3$$ inches?

Accepted Answer

$$4.05$$ pounds

Question 18

Find the constant of variation $$k$$ and the variation equation if $$y$$ varies jointly as $$x$$ and the cube of $$z$$, and $$y = 120$$ when $$x = 5$$ and $$z = 2.$$

Accepted Answer

$$k = 3$$, $$y = 3xz^3$

Question 19

The maximum weight $$W a $$rectangular beam can support varies jointly as its width $$w$$ and the square of its height $$h$$, and inversely as its length $$l. If a $$beam $$\dfrac{1}{2} ft $$wide, $$\dfrac{1}{3} ft $$high, and $$10 ft $$long can support $$12$$ tons, how much can a similar beam $$\dfrac{2}{3} ft $$wide, $$\dfrac{1}{2} ft $$high, and $$16 ft $$long support?

Accepted Answer

$$8$$ tons

Rational Expressions and Functions: Operations, Applications, and Variation

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