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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. 65 (page 242)

In Problems 63鈥�66, (a) graph each function by hand, and (b) solve $f (x) 鈮� 0 .$
$f (x) = \frac{x^{3} + 2 x^{2} - 11 x - 12}{x^{2} - x - 6}$

Short Answer

Expert verified

Part a. The graph of the function is

Part b. The solution of $f (x) 鈮� 0$ is $(- 4 鈮� x < - 2 a n d - 1 鈮� x < 鈭�) - {3} .$

Step by step solution

01

Part (a) Step 1. Given Information

The given function is $f (x) = \frac{x^{3} + 2 x^{2} - 11 x - 12}{x^{2} - x - 6}$

We have to graph the function.

02

Part (a) Step 2. Sketch the graph of the function

The graph of the function is

03

Part (b) Step 1. Given Information

The given function is $f (x) = \frac{x^{3} + 2 x^{2} - 11 x - 12}{x^{2} - x - 6}$

We have to solve for $f (x) 鈮� 0$

04

Part (b) Step 2. Solving for f(x)≥0

So First, case:

$\frac{x^{3} + 2 x^{2} - 11 x - 12}{x^{2} - x - 6} 鈮� 0 x^{3} + 2 x^{2} - 11 x - 12 鈮� 0 a n d x^{2} - x - 6 > 0 (x + 1) (x + 4) (x - 3) 鈮� 0 (x - 3) (x + 2) > 0 - 4 鈮� x 鈮� - 1 a n d x 鈮� 3 x > 3 a n d x < - 2 [C o m m o n s o l u t i o n] x > 3 a n d - 4 鈮� x < - 2$

Second case:

$\frac{x^{3} + 2 x^{2} - 11 x - 12}{x^{2} - x - 6} 鈮� 0 x^{3} + 2 x^{2} - 11 x - 12 鈮� 0 a n d x^{2} - x - 6 > 0 (x + 1) (x + 4) (x - 3) 鈮� 0 (x - 3) (x + 2) > 0 - 1 鈮� x 鈮� 3 a n d x 鈮� - 4 - 2 < x < 3 [C o m m o n s o l u t i o n] - 1 鈮� x < 3$

Thus, the solution is $(- 4 鈮� x < - 2 a n d - 1 鈮� x < 鈭�) - {3} .$

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