Problem 73 - College Algebra

Question

For the following exercises, use a calculator to graph \(f(x) .\) Use the graph to solve \(f(x)>0\) $$ f(x)=\frac{x+2}{(x-1)(x-4)} $$

Step-by-Step Solution

Answer

The solution to \( f(x) > 0 \) is \((-2, 1)\) and \((4, \infty)\).

Step 1: Identify the function components

The function is given by \( f(x) = \frac{x+2}{(x-1)(x-4)} \). The numerator is \( x + 2 \), a linear expression. The denominator is \( (x-1)(x-4) \), a quadratic expression with roots at \( x = 1 \) and \( x = 4 \). These roots will affect the regions where \( f(x) \) is positive or negative.

Step 2: Analyze the sign changes on the number line

Determine the critical points for sign changes, which are \( x = -2 \), \( x = 1 \), and \( x = 4 \). Test the sign of \( f(x) \) in each interval: \((-\infty, -2)\), \((-2, 1)\), \((1, 4)\), and \((4, \infty)\). Choose test points from each interval to determine where \( f(x) > 0 \).

Step 3: Evaluate test points

- Interval \((-\infty, -2)\): Choose \( x = -3 \), then \( f(-3) = \frac{-1}{15} < 0 \).- Interval \((-2, 1)\): Choose \( x = 0 \), then \( f(0) = \frac{2}{4} = 0.5 > 0 \).- Interval \((1,4)\): \( f(x) \) is undefined at \( x = 1 \), check \( x = 2 \), then \( f(2) = -0.33 < 0 \).- Interval \((4, \infty)\): Choose \( x = 5 \), then \( f(5) = \frac{3}{15} = 0.2 > 0 \).

Step 4: Determine solution

Based on test points, \( f(x) > 0 \) in intervals \((-2, 1)\) and \((4, \infty)\). At \( x = 1 \) and \( x = 4 \), \( f(x) \) is undefined due to division by zero, so these points are not included in the solution.

Key Concepts

Graphing Functions

Graphing functions, especially rational ones, is crucial for understanding their behavior.
The function given here, a rational function, is expressed as the quotient of two polynomials.

How to graph:

• Identify both the numerator and the denominator.
• The roots of the denominator indicate potential asymptotes, pointing to values of x where the function is undefined.
• Use a calculator or software to input the function and obtain a visual graph, which displays all critical points.

In this case, graphing helps determine function behavior across different x-values, allowing for a clearer understanding of when the function remains positive.

Inequalities in Algebra

Algebraic inequalities allow us to explore when a function is positive (or negative).
For rational functions like ours, setting up the inequality \( f(x) > 0 \) examines which parts of the graph lie above the x-axis.

Steps to solve inequalities:

• Identify critical points where the function changes sign. These include solving the numerator and denominator separately.
• Create intervals between these critical points on a number line.
• Test sample values from each interval to determine the function's sign.

By following these steps, students can determine which intervals overlap with periods of positivity or negativity, offering insight into real-world applications.

Function Positivity

Determining when a function is positive involves identifying intervals where the function's output, \( f(x) \), remains above zero, indicating positive regions on the graph.
This is indispensable for analyzing functions across their domains
For our rational function:

• Examine critical points, especially where \( f(x) \) changes sign.
• Consider the intervals identified as positive: \((-2, 1)\) and \((4, \infty)\).
• Recognize undefined points due to division by zero, which do not contribute.

By understanding which intervals are positive, we derive insight into its practical significance and predict how the function behaves over different values.

Asymptotes

Asymptotes are vital in understanding rational functions, providing insights into the behavior of graphs as they approach undefined regions.
For the given rational function, vertical asymptotes occur at the roots of the denominator, where the function is undefined.

Types of asymptotes:

• Vertical Asymptotes: Occur at \( x = 1 \) and \( x = 4 \). These result from the denominator becoming zero, leading to undefined function values.
• Horizontal Asymptotes: Often analyzed by long-term behavior. They describe how a function behaves as \( x \to \infty \) or \( x \to -\infty \).

Recognizing these asymptotes allows for prediction of function behavior near boundaries, and how it aligns with core concepts like positivity and sign changes across intervals.