Problem 525 - Precalculus

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: Analyze the function

Recognize that the function given is a rational function, specifically a reciprocal function of the form \( f(x) = \frac{4}{2x - 3} \). This function will have a vertical asymptote where the denominator equals zero and a horizontal asymptote along the x-axis.

Step 2: Identify the vertical asymptote

To find the vertical asymptote, set the denominator equal to zero: \[ 2x - 3 = 0 \]Solving this equation gives:\[ x = \frac{3}{2} \]So, the vertical asymptote is at \( x = \frac{3}{2} \).

Step 3: Determine the function behavior

Before graphing, consider how the function behaves near the vertical asymptote. For \( x < \frac{3}{2} \), the function value is negative, and for \( x > \frac{3}{2} \), it is positive.

Step 4: Graph the function using a calculator

Enter the function \( f(x) = \frac{4}{2x - 3} \) into a graphing calculator. Observe the graph for the vertical asymptote at \( x = \frac{3}{2} \) and note where the curve is above the x-axis (positive values).

Step 5: Interpret the graph for \( f(x) > 0 \)

From the graph, \( f(x) \) is greater than zero for all \( x > \frac{3}{2} \). The rational function is discontinuous at \( x = \frac{3}{2} \) due to the vertical asymptote.

Key Concepts

Vertical Asymptote Understanding

A vertical asymptote is a line where a graph approaches but never quite touches or crosses. For rational functions, vertical asymptotes occur where the denominator equals zero. In our example, the function is \( f(x) = \frac{4}{2x - 3} \).
To find the vertical asymptote, we set the denominator \(2x - 3\) to zero because this is where the function "blows up" or becomes undefined:
\[ 2x - 3 = 0 \]
Solving this simple algebraic equation results in:
\[ x = \frac{3}{2} \]
This means that at \( x = \frac{3}{2} \), the graph of \( f(x) \) will have a vertical asymptote.
An important thing to note is the behavior of the graph near a vertical asymptote. As \( x \) approaches \( \frac{3}{2} \) from the left, the graph will trend towards negative infinity, and as it approaches from the right, it trends towards positive infinity.

Reciprocal Function Simplified

A reciprocal function is a type of rational function represented by \( f(x) = \frac{k}{x-a} \), where \(k\) is a constant and \(x - a\) forms the denominator.
For our function, \( f(x) = \frac{4}{2x - 3} \), the constant \(k\) is 4, and the function denominator is \(2x - 3\).
This puts us in a typical reciprocal function format with \(a = \frac{3}{2}\).
Reciprocal functions have important characteristics such as vertical asymptotes, which we've found at \(x = \frac{3}{2}\).
Additionally, such functions may also have horizontal asymptotes, often determined by comparing the degrees of numerator and denominator.
For this specific case, since the numerator is a constant and the denominator a simple linear term, the horizontal asymptote is the x-axis.

Using a Graphing Calculator

Graphing calculators are powerful tools that help visualize complex functions like reciprocal functions.
To graph \( f(x) = \frac{4}{2x - 3} \), you’ll need a graphical calculator that can input rational expressions.
Simply enter the function exactly as displayed in the calculator’s function input area.

• Observe how the graph approaches the vertical line at \(x = \frac{3}{2}\).
• Your graph will show a curve in different directions around this line.
• You can visually understand the behavior of \(f(x)\) around this critical point well.
• The graph will be in two sections: decreasing towards the vertical asymptote from the left and increasing from the right.

When studying this graph, note where the graph is above or below the x-axis, which aligns with the function's positivity and negativity discussed earlier.
This visual will confirm that \( f(x) > 0 \) for \( x > \frac{3}{2} \). The descriptive power of graphing calculators makes seeing mathematical concerns like asymptotic behavior straightforward.