Problem 10 For the function $$ g(x)=\fr... [FREE SOLUTION] (2024)

Chapter 4: Problem 10

For the function $$ g(x)=\frac{x-2}{x+4} $$ solve each of the following. $$g(x) < 0$$

Short Answer

Expert verified

$ g(x) < 0 $ for $ x \in (-4, 2) $

Step by step solution

Determine the critical points

Identify the points where the function $ g(x) $ equals zero or where the function is undefined. Set $ g(x) = 0 $: $ \frac{x-2}{x+4} = 0 $ implies $ x - 2 = 0 $. Thus, $ x = 2 $ is a critical point. The function is undefined when the denominator is zero, so set $ x + 4 = 0 $ to find $ x = -4 $. Therefore, the critical points are $ x = 2 $ and $ x = -4 $.

Divide the number line

Divide the number line into intervals based on the critical points: $ (-\infty, -4) $, $ (-4, 2) $, and $ (2, \infty) $. These intervals are created by the points where the function changes its behavior.

Test each interval

Choose a test point from each interval and substitute it into $ g(x) $ to determine if $ g(x) < 0 $. For the interval $ (-\infty, -4) $, choose $ x = -5 $: $ g(-5) = \frac{-5 - 2}{-5 + 4} = \frac{-7}{-1} = 7 $ (positive, so this interval does not satisfy $ g(x) < 0 $). For the interval $ (-4, 2) $, choose $ x = 0 $: $ g(0) = \frac{0 - 2}{0 + 4} = \frac{-2}{4} = -0.5 $ (negative, so this interval satisfies $ g(x) < 0 $). For the interval $ (2, \infty) $, choose $ x = 3 $: $ g(3) = \frac{3 - 2}{3 + 4} = \frac{1}{7} $ (positive, so this interval does not satisfy $ g(x) < 0 $).

Combine the results

Since the function is negative only in the interval $ (-4, 2) $, this is the solution to the inequality $ g(x) < 0 $.

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Critical Points

Critical points are values of x where the function changes its behavior in specific ways. These points can be where the function equals zero or where it is undefined. For the function given in the problem, let's find these points by:

Setting the numerator equal to zero: For the function $ g(x) = \frac{x-2}{x+4} $, set $ x-2 = 0 $. This gives us $ x = 2 $.
Finding where the denominator is zero: Set $ x+4 = 0 $. This gives $ x = -4 $.

By doing this, we identified the critical points as $ x = 2 $ and $ x = -4 $. These points help us understand where the function can potentially switch from positive to negative values or vice versa.

Interval Testing

Interval testing involves dividing the number line into intervals based on the critical points and determining the sign of the function within each interval. Let's break it down with the example function:

First, create intervals based on critical points $ x = 2 $ and $ x = -4 $: $ (-\infty, -4) $, $ (-4, 2) $, and $ (2, \infty) $.
Select a test point from each interval and substitute it into the function to check whether the function is positive or negative in that particular interval.

For the interval $ (-\infty, -4) $, choose $ x = -5 $. When substituted, $ g(-5) = \frac{-7}{-1} = 7 $ (positive).
For the interval $ (-4, 2) $, choose $ x = 0 $. When substituted, $ g(0) = \frac{-2}{4} = -0.5 $ (negative).
For the interval $ (2, \infty) $, choose $ x = 3 $. When substituted, $ g(3) = \frac{1}{7} $ (positive).
Only the interval $ (-4, 2) $ has a negative value, which meets the criteria of $ g(x) < 0 $.

Number Line Analysis

Number line analysis helps visualize where the function changes its sign. Once we determine the critical points and perform interval testing, we can represent our findings on a number line. Here's how:

Draw a horizontal line representing all possible values of $ x $.
Mark critical points $ x = -4 $ and $ x = 2 $ on this line.
Check the sign of the function within each interval you previously analyzed: $ (-\infty, -4) $, $ (-4, 2) $, $ (2, \infty) $.

Mark the intervals that meet the condition $ g(x) < 0 $ by shading or highlighting them. Since the function is negative in the interval $ (-4, 2) $, this interval would be shaded or highlighted, showing that $ g(x) < 0 $ only within this range.

Rational Functions

Rational functions are functions expressed as the quotient of two polynomials. In this problem, we considered the rational function $ g(x) = \frac{x-2}{x+4} $. Here's what you need to know about them:

They can have vertical asymptotes where the denominator equals zero (e.g., at $ x = -4 $).
They may have horizontal asymptotes based on the degrees of the numerator and denominator polynomials.
Evaluating their behavior over different intervals can help determine where the function is positive or negative.

By understanding these characteristics, we effectively solved the inequality $ g(x) < 0 $ by identifying critical points, testing intervals, and using number line analysis. This approach allows us to grasp the behavior of rational functions and how they change over different intervals.

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