Problem 71 - College Algebra

Question

For the following exercises, use a calculator to graph \(f(x) .\) Use the graph to solve \(f(x)>0\) $$ f(x)=\frac{4}{2 x-3} $$

Step-by-Step Solution

Answer

\( f(x) > 0 \) for \( x > \frac{3}{2} \).

Step 1: Identify the Domain of f(x)

The function is defined as \( f(x) = \frac{4}{2x-3} \). The denominator cannot be zero, so solve \( 2x-3 = 0 \) for \( x \). This gives \( x = \frac{3}{2} \). Therefore, the domain of \( f(x) \) is all real numbers except \( x = \frac{3}{2} \).

Step 2: Determine Asymptotes

Identify vertical and horizontal asymptotes. The vertical asymptote is at \( x = \frac{3}{2} \) because the function is undefined there. To find the horizontal asymptote, as \( x \to \infty \) or \( x \to -\infty \), \( f(x) = \frac{4}{2x-3} \to 0 \). Thus, the horizontal asymptote is \( y = 0 \).

Step 3: Sketch the Graph

Using a calculator, graph \( f(x) = \frac{4}{2x-3} \). The graph will have a vertical asymptote at \( x = \frac{3}{2} \) and approach the horizontal asymptote \( y = 0 \). It will exist in the regions where \( x < \frac{3}{2} \) and \( x > \frac{3}{2} \).

Step 4: Determine Where f(x) > 0

From the graph, identify where the function is above the x-axis. For \( x < \frac{3}{2} \), \( f(x) \) is negative (below the x-axis). For \( x > \frac{3}{2} \), \( f(x) \) is positive (above the x-axis).

Step 5: Conclusion

From steps 2 to 4, conclude that \( f(x) > 0 \) for \( x > \frac{3}{2} \).

Key Concepts

Domain of a Function

When it comes to rational functions like \( f(x) = \frac{4}{2x-3} \), understanding the domain is crucial. The domain refers to all the possible input values \( x \) for which the function is defined. Since division by zero is undefined, we need to identify values of \( x \) that make the denominator zero and exclude them from the domain.
To find these values, set the denominator equal to zero and solve for \( x \): \( 2x - 3 = 0 \). Solving this equation gives \( x = \frac{3}{2} \). Therefore, all real numbers are in the domain except \( x = \frac{3}{2} \).
In simpler terms:

• Look for values that zero-out the denominator.
• Exclude these values from the domain.
• State the domain based on all valid \( x \) values.

This ensures your function keeps its mathematical integrity and remains continuous wherever it's defined.

Asymptotes

Asymptotes are lines that a graph approaches but never actually meets. They tell us how a function behaves at its extremes and where it might "shoot off" toward infinity. For\( f(x) = \frac{4}{2x-3} \), there are two key types of asymptotes to identify: vertical and horizontal.
First, the vertical asymptote occurs where the function is undefined, which happens at \( x = \frac{3}{2} \) in this case. This asymptotic line flags a "break" in the graph where it abruptly ascends or descends.
Next, consider the horizontal asymptote, which shows us the behavior of \( f(x) \) as \( x \) becomes indefinitely larger or smaller. For this function, as \( x \to \infty \) or \( x \to -\infty \), \( f(x) \) approaches \( 0 \). Thus, the horizontal asymptote here is \( y = 0 \).
In brief, identify asymptotes by:

• Finding denominator zeros for vertical asymptotes.
• Observing the function's end behavior for horizontal asymptotes.

This understanding is vital when sketching graphs and predicting the path of rational functions.

Analyzing Sign of Functions

To determine where a function like \( f(x) = \frac{4}{2x-3} \) is positive, or \( f(x) > 0 \), analyzing its graph is very helpful. The sign of the function varies depending on whether it is above or below the x-axis.
First, look at each interval created by the detected vertical asymptote in the domain. For \( x < \frac{3}{2} \), the function yields negative values, appearing below the x-axis. Conversely, for \( x > \frac{3}{2} \), the function's values are positive, placing it above the x-axis.
Graphically:

• Check quantities on intervals determined by vertical asymptotes.
• Identify which intervals yield positive or negative values.
• Concisely state intervals where the function is positive.

In this case, \( f(x) > 0 \) for \( x > \frac{3}{2} \). This helps quickly predict and verify the behavior of rational functions across different regions of their domain.