Q: Which of the following is the correct domain for the rational function $$f(x) = \frac{x^2$ + 3}{2x^2$ + 4x - 5}$$?

$$\{x \mid x \neq 1, x \neq -2.5\}$$

Question 1

Which of the following is the correct domain for the rational function $$f(x) = \frac{x^2$ + 3}{2x^2$ + 4x - 5}$$?

Accepted Answer

$$\{x \mid x 
eq 1, x 
eq -2.5\}$$

Question 2

Given the rational function $$f(x) = \frac{x+1}{x-2}$$, what happens to $$f(x) as$$ $$x$$ approaches 2 from the left?

Accepted Answer

$$f(x)$$ approaches -\infty$

Question 3

A company’s average cost per item to produce $$x$$ pairs of earbuds is $$C(x) = \frac{12x + 6000}{x}. If $$the company expects to sell 2000 pairs, what is the average cost per item?

Accepted Answer

$$15$$

Question 4

Simplify the expression $$\frac{x^2$ + 3x + 2}{x^2$ - 1}.$$

Accepted Answer

$$\frac{x+2}{x-1}$$

Question 5

If $$y$$ varies directly as $$x$$ and $$y = 48$$ when $$x = 12$$, what is the equation relating $$y$$ and $$x$$?

Accepted Answer

$$y = 4x$$

Question 6

Solve the rational equation $$\frac{3x}{2x-1} = 3.$$

Accepted Answer

$$x = 1$$

Question 7

The stopping distance $$d of a $$car after the brakes are applied varies directly as the square of the speed $$r. If$$ $$d = 50 ft $$when $$r = 30$$ mph, what is $$d$$ when $$r = 36$$ mph?

Accepted Answer

$$72 ft$$

Question 8

Which of the following is the reciprocal of $$\frac{2x-1}{x+3}$$?

Accepted Answer

$$\frac{x+3}{2x-1}$$

Question 9

A pump can empty a swimming pool in 50 hours. A second pump can empty it in 80 hours. How long will it take both pumps working together to empty the pool?

Accepted Answer

$$\frac{400}{13}$$ hours

Question 10

Simplify the complex fraction $$\frac{1+2x}{1-2x}.$$

Accepted Answer

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

Question 11

The area $$A of a $$rectangle is $$3x^2 + 14x + 15$ and its width is x + 3$. What is the length L$ of the rectangle in terms of x$?

Accepted Answer

$$3x + 5$$

Question 12

Find the least common multiple (LCM) of $$x^2 + 3x + 2$ and x^2$ - x - 2.$$

Accepted Answer

$$(x+1)(x^2-4$$)$$

Question 13

If $$y$$ varies inversely as $$x$$ and $$y = 16$$ when $$x = 0.3$$, what is the variation constant $$k$$?

Accepted Answer

$$k = 4.8$$

Question 14

Divide $$5x^3$$ - $$15x^2 by$$ $$5x^2.$$

Accepted Answer

$$x - 3$$

Question 15

The volume $$V of $$wood in a tree varies jointly as the height $$h$$ and the square of the girth $$g. If$$ $$V = 216 m$^3$$ when $$h = 30 m $$and $$g = 1.5 m$$, what is the value of the constant $$k in$$ $$V = k h g^2$?

Accepted Answer

$$k = 3.2$$

Question 16

Which of the following best describes the graph of $$y = \frac{k}{x}$$ for $$k \u003e 0$$?

Accepted Answer

As $$x$$ increases, $$y$$ decreases

Intermediate Algebra: Rational Expressions, Equations, and Variation

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