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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 4: Q. 11 (page 234)

In Problems $7 - 44$ , follow Steps $1$ through $7$ on page $228$ to analyze the graph of each function.
$R (x) = \frac{3}{x^{2} - 4}$

Short Answer

Expert verified

The graph for the function $R (x) = \frac{3}{x^{2} - 4}$ is,

Step by step solution

01

Step 1 Given function is,

$R (x) = \frac{3}{x^{2} - 4}$

which may also written as,

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

The domain of $R$ is, $\{x | x 鈮� 卤 2\}$

As $0$ is in the domain, substitute $x = 0$ in the given function.

$R (x) = \frac{3}{(+ 2) (- 2)} = \frac{- 3}{4}$

So, the obtained point is, $(0, \frac{- 3}{4})$ .

02

Step 2 Ris in the lowest terms.

The real zeros of the denominator are the real solutions of the equation.

$(x + 2) (x - 2) = 0 x = 卤 2$

The two lines $x = 2, x = - 2$ are the two vertical asymptotes of the graph.

In the given function, degree of the numerator is less than the degree of the denominator. So, $R$ is proper.

So, the line $y = 0$ is the horizontal asymptote of the graph.

Substitute $R (x) = 0$ in the given function.

$0 = \frac{3}{x^{2} - 4} 0 鈮� 3$

There exists no solution. So, the graph of $R$ does not intersect the horizontal asymptote at $y = 0$ .

03

Step 3 Now plot a graph by using the known values.

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Most popular questions from this chapter

Find the vertical, horizontal, and oblique asymptotes, if any, of each rational function.

$G (x) = \frac{x^{4} - 1}{x^{2} - x}$

If a rational function is proper, then______ is a horizontal

asymptote.

In Problems 13鈥�24, find the domain of each rational function.

$R (x) = \frac{3 (x^{2} - x - 6)}{4 (x^{2} - 9)}$

Solve the inequality algebraically.

$x^{4} > x^{2}$

Find the real zeros of f. Use the real zeros to factor f.

$f (x) = 4 x^{4} - 15 x^{3} - 8 x^{2} + 15 x + 4$

See all solutions

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