Question 1

For which value of $$x is $$the rational expression $$\frac{2x+5}{x^2$-9}$$ undefined?

Accepted Answer

$$x = 3$$

Question 2

Simplify the expression $$\frac{x^2$-4}{x^2$+2x}$$ divided by $$\frac{x-2}{x}.$$

Accepted Answer

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

Question 3

Subtract and simplify: $$\frac{3}{x+1} - \frac{2}{x-1}.$$

Accepted Answer

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

Question 4

Simplify the complex fraction $$\frac{\frac{1}{x} + \frac{2}{y}}{\frac{3}{x}}.$$

Accepted Answer

Question 5

Given $$f(x) = x^2$ - 2x$$, find the average rate of change of $$f on $$the interval $$[1, 4].$$

Accepted Answer

$$6$$

Question 6

Rationalize the denominator: $$\frac{5}{\sqrt{3}}.$$

Accepted Answer

$$\frac{5\sqrt{3}}{3}$$

Question 7

Convert $$\sqrt[3]{x^5$} to $$rational exponent form.

Accepted Answer

$$x^{5/3}$$

Question 8

Simplify $$\sqrt{x^4$y^2$$}$$ assuming x$$, y \geq 0$.

Accepted Answer

$$x^2y$

Question 9

Perform and simplify: $$\sqrt{18} + 2\sqrt{8}.$$

Accepted Answer

$$3\sqrt{2} + 4\sqrt{2} = 7\sqrt{2}$$

Question 10

State the horizontal asymptote of the rational function $$f(x) = \frac{2x^2$+1}{x^2$-3}.$$

Accepted Answer

$$y = 2$$

Question 11

Find the vertical asymptotes and holes of $$f(x) = \frac{x^2$-4}{x^2$-5x+6}.$$

Accepted Answer

Vertical asymptotes at $$x=2$$ and $$x=3$$; no holes.

Question 12

Find the oblique asymptote of $$f(x) = \frac{x^2$+3x+2}{x+1}.$$

Accepted Answer

$$y = x + 2$$

Question 13

Using the table below, estimate $$\lim_{x 	o 2} f(x)$$ where $$f(x)$$ approaches the following values as $$x$$ gets closer to $$2$$: $$f(1.9) = 3.8$$, $$f(1.99) = 3.98$$, $$f(2.01) = 4.02$$, $$f(2.1) = 4.2.$$

Accepted Answer

$$4$$

Question 14

Evaluate $$\lim_{x 	o 0} \frac{\sin x}{x}$$ using limit properties.

Accepted Answer

$$1$$

Question 15

Solve the inequality $$x^2 - 5x + 6$$ \u003c 0$$ using test points.

Accepted Answer

$$2 \u003c x \u003c 3$$

Question 16

Given the graph of $$f(x) = \frac{1}{x-2}$$, for which values of $$x is$$ $$f(x) \u003e 0$$?

Accepted Answer

$$x \u003e 2$$

Question 17

A medication's concentration in the bloodstream is modeled by $$C(t) = \frac{10}{t+2}$$, where $$t is $$time in hours. What is the long-term concentration as $$t 	o \infty$$?

Accepted Answer

$$0$$

Question 18

Simplify $$\sqrt{50} - \sqrt{18}.$$

Accepted Answer

$$5\sqrt{2} - 3\sqrt{2} = 2\sqrt{2}$$

Question 19

Convert $$x^{-3/4} to $$radical form.

Accepted Answer

$$\frac{1}{\sqrt[4]{x^3$}}$$

Question 20

A chemical reaction rate is modeled by $$R(x) = \frac{5x}{x^2$+1}. $$Find the horizontal asymptote of $$R(x).$$

Accepted Answer

$$y = 0$$

Precalculus Test 1 Study Guide: Rational Expressions, Radicals, Asymptotes, Limits, and Inequalities

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