Problem 70 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

The solution is \( x > -1 \).

Step 1: Identify the function

The function given is \( f(x) = \frac{2}{x+1} \). We will analyze this function to understand its behavior and find the values of \( x \) for which \( f(x) > 0 \).

Step 2: Determine the asymptote

For the function \( f(x) = \frac{2}{x+1} \), there is a vertical asymptote at \( x = -1 \) because the denominator becomes zero at this value, making the function undefined.

Step 3: Consider the sign of the function

Since the numerator 2 is positive, the sign of \( f(x) \) depends on the denominator \( x+1 \). \( f(x) > 0 \) when \( x+1 > 0 \), which simplifies to \( x > -1 \).

Step 4: Graph the function

Use a graphing calculator to plot \( f(x) = \frac{2}{x+1} \). The graph will show that to the right of the vertical asymptote at \( x = -1 \), the function values are positive.

Step 5: Solution interpretation

Based on the graph and analysis, \( f(x) > 0 \) for all \( x > -1 \). This is because, as \( x \) exceeds \( -1 \), the denominator is positive, making the whole fraction positive.

Key Concepts

Vertical Asymptote

In the function \( f(x) = \frac{2}{x+1} \), the vertical asymptote plays a crucial role in understanding the behavior of the function. A vertical asymptote occurs where the function is undefined. This generally happens at points where the denominator equals zero. Here, the denominator \( x + 1 \) is zero when \( x = -1 \).

This means that the function grows without bounds as we approach \( x = -1 \). On the graph, lines representing vertical asymptotes are often shown as dashed, signaling a barrier that the function cannot cross. Understanding vertical asymptotes is vital for graphing rational functions because they highlight divisions within the graph that separate different behaviors and trends in the function. When interpreting rational functions, always start by identifying these asymptotes.

Function Behavior Analysis

Analyzing the behavior of the function \( f(x) = \frac{2}{x+1} \) can help us determine where the function is positive or negative. Given the numerator of the function is always positive, the sign of the whole function depends on the denominator \( x + 1 \).

- The function is undefined at \( x = -1 \) due to the zero in the denominator.
- When \( x > -1 \), \( x + 1 \) is positive, hence \( f(x) \) can be positive as the positive numerator is divided by a positive denominator.
- Conversely, when \( x < -1 \), \( f(x) \) will be negative, since the denominator becomes negative while the numerator remains positive.

By studying these intervals, we can predict the sign of \( f(x) \), making it easier to graph and analyze. Such understanding aids in resolving inequalities involving rational functions, a key step in many algebraic problems.

Graph Interpretation

When analyzing the graph of \( f(x) = \frac{2}{x+1} \), interpretation is enhanced by recognizing key features: the vertical asymptote and sections where the graph is above or below the x-axis. The graph visually represents the function's behavior in relation to its asymptote and overall positivity.

After plotting \( f(x) \) on a calculator, you will observe that:

• To the right of the vertical asymptote at \( x = -1 \), the function values remain positive, indicating \( f(x) > 0 \).
• As you move left past the asymptote, the function dips below the x-axis, making \( f(x) < 0 \).

Recognizing these sections on a graph helps in solving inequalities and offers insights into the function's real-world applications, such as physical systems that behave similarly. Becoming proficient at quickly interpreting graphs unlocks a deeper understanding of complex mathematical concepts.