Problem 71 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

The solution is \( x < \frac{3}{2} \) and \( x > 2 \).

Step 1: Graph the Function

Begin by entering the function \( f(x) = \frac{4}{2x - 3} \) into a graphing calculator. Look at the graph carefully and observe its general shape, which will be a hyperbola. Identify any asymptotes or intercepts on the graph.

Step 2: Identify the Asymptotes

Determine the vertical asymptote by setting the denominator equal to zero: \( 2x - 3 = 0 \). Solving for \( x \) gives \( x = \frac{3}{2} \). This is where the graph will have a vertical asymptote. Note that there is no horizontal asymptote in this kind of function.

Step 3: Locate the Regions Where the Function is Positive

Examine the graph to see where \( f(x) \) is above the x-axis. Typically, this will occur in two distinct regions separated by the vertical asymptote. Trace these regions to identify for which values of \( x \), the output \( f(x) \) is greater than zero.

Step 4: Determine Specific Intervals

From observing the graph, identify the intervals where the function \( f(x) \) is positive. This should be in the interval \( x < \frac{3}{2} \) and \( x > 2 \). This means the graph is above the x-axis in these regions.

Key Concepts

Using a Calculator

To begin graphing rational functions, like the one given in the exercise, using a calculator can be incredibly helpful. Modern calculators are equipped with graphing capabilities that allow you to visualize complex functions quickly. To start, you will need to enter the function\[ f(x) = \frac{4}{2x - 3} \]into your calculator's graphing mode. Ensure that you input the function correctly to avoid errors.
Once entered, observe the plotted graph. Pay special attention to shapes and intersections. Besides offering visual aid, the calculator can numerically evaluate specific points and give you a perspective on how the function behaves for various values of \(x\). Using such technology enhances your capability to solve more complex equations with better precision.

• Enter the function accurately to avoid misinterpretation.
• Zoom in or adjust the view if necessary to get a clearer picture of details.
• Note down important features like zeroes, intercepts, or asymptotic behavior.

Hyperbola

A hyperbola is the graphical representation of rational functions like the one in this exercise. Specifically, it appears when graphing a function of the form\[ f(x) = \frac{a}{bx + c} \].
This distinct shape generally has two curved branches extending in opposite directions, reflecting the symmetrical property of hyperbolas.
The two branches of the hyperbola are separated by asymptotes, which are lines that the graph approaches but never touches or crosses. Understanding hyperbolas is crucial, as they reveal significant characteristics about the function itself.

• Composed of two symmetric open curves.
• Always feature asymptotic behavior towards axes or specific lines.
• They may never intersect with their asymptotes.

Overall, recognizing hyperbolas as a shape is essential when interpreting rational function behaviors.

Vertical Asymptote

Vertical asymptotes are crucial lines you encounter often when graphing rational functions. For the function given,\[ f(x) = \frac{4}{2x - 3} \],
the vertical asymptote occurs where the denominator is zero. This is because division by zero is undefined, causing the function to be unbounded. To find the vertical asymptote, solve for when\[ 2x - 3 = 0 \].
Doing the math gives\[ x = \frac{3}{2} \].
On a graph, this is where the hyperbola will appear to 'split' upwards and downwards, creating two separate branches.

• Occurs at undefined points of the function.
• Dividing by zero results in vertical asymptotes.
• Graphs approach but never intersect vertical asymptotes.

This concept further aids in understanding how and where rational functions increase or decrease.

Solving Inequalities

Rational functions often require you to determine where they are positive or negative, i.e., solving inequalities like \( f(x) > 0 \).
For the function\[ f(x) = \frac{4}{2x - 3} \],
you aim to find regions where the graph is above the x-axis. Thank the calculator for this, as it visually demonstrates positive intervals for you. By identifying where the graph is positioned above the x-axis, you locate intervals that satisfy the inequality. In this case, intervals such as \( x < \frac{3}{2} \) and \( x > 2 \) result in positive values for \( f(x) \).

• Inequalities reveal parts of the graph above or below the x-axis.
• Graphs help visualizing solutions of these inequalities.
• Stay mindful of asymptotes; they split the graph into distinct parts.

Mastering this concept eases the process of analyzing where functions behave in desired ways, critical for various real-world applications.