Chapter 4: Q. 65 (page 242)

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

Short Answer

Expert verified

Part a. The graph of the function is

Part b. The solution of $f (x) \geq 0$ is $(- 4 \leq x < - 2 a n d - 1 \leq x < \infty) - {3} .$

Step by step solution

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.

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

The graph of the function is

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 $f (x) \geq 0$

Part (b) Step 2. Solving for f(x)≥0

So First, case:

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

Second case:

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

Thus, the solution is $(- 4 \leq x < - 2 a n d - 1 \leq x < \infty) - {3} .$

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

Short Answer

Step by step solution

Part (a) Step 1. Given Information

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

Part (b) Step 1. Given Information

Part (b) Step 2. Solving for f(x)≥0

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In Problems 63–66, (a) graph each function by hand, and (b) solve f(x)≥0.f(x)=x3+2x2-11x-12x2-x-6

Short Answer

Step by step solution

Part (a) Step 1. Given Information

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

Part (b) Step 1. Given Information

Part (b) Step 2. Solving for f(x)≥0

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Calculus

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Probability and Statistics

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Applied Mathematics

Study anywhere. Anytime. Across all devices.

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