Problem 84 - College Algebra Essentials

Question

Use a graphing utility to graph \(f\) and \(\operatorname{gin}\) the same viewing rectangle. In addition, graph the line \(y=x\) and visually determine if \(f\) and \(g\) are inverses. $$f(x)=4 x+4, g(x)=0.25 x-1$$

Step-by-Step Solution

Answer

The function \(f(x)=4x+4\) and its proposed inverse \(g(x)=0.25x-1\) are indeed mirror reflections around the line \(y=x\). Thus, \(f\) and \(g\) are inverse functions.

Step 1: Graph the given functions

Using a graphing utility, plot the functions \(f(x)=4x+4\), \(g(x)=0.25x-1\), and the line \(y=x\). Both the functions and the line will appear on the same graph.

Step 2: Identify if \(f\) and \(g\) are reflections of \(y=x\)

The visual benchmark for judging if two functions are inverses is if they are mirror images across the line \(y=x\). Therefore, observe the plot and identify if \(f\) and \(g\) are symmetrical about the line \(y=x\).

Step 3: Visual interpretation

If \(f\) and \(g\) are mirror reflections of each other along the \(y=x\) line, these two functions are inverses. If not, they're not inverses.

Key Concepts

Graphing Utility

A graphing utility is a powerful tool used to visually represent functions. These can either be software applications on calculators or computer programs.
By plotting graphs using a graphing utility, you save time and avoid manual plotting errors.
For this exercise, you need to use a graphing utility to graph the functions \( f(x) = 4x + 4 \), \( g(x) = 0.25x - 1 \), and the line \( y = x \) in a single viewing rectangle.
This means all these graphs will be on the same coordinate plane, making it easy to compare their shapes and positions.
To use a graphing utility:

• Input the equation of each function
• Select a range for the x-axis and y-axis that reveals the effects of each function (ensure it's large enough to capture the behavior of each function)
• Visually inspect the graph for intersections, symmetry, and patterns

By using this method, you will be able to quickly assess mathematical questions related to graph symmetry, intersections, and other functional relations.

Linear Functions

Linear functions are equations of the form \( f(x) = ax + b \) where \( a \) and \( b \) are constants. These functions yield straight lines when graphed.

In the exercise, \( f(x) = 4x + 4 \) and \( g(x) = 0.25x - 1 \) are both linear functions. Let's break down these functions:

• The slope \( a \) of a linear function determines the tilt or steepness of the line. A higher slope means a steeper line.
• The y-intercept \( b \) is the point where the line crosses the y-axis. This helps position the line vertically on the graph.

For \( f(x) \):

• Slope \( a = 4 \), meaning the line is steeper.
• Y-intercept \( b = 4 \), so it crosses the y-axis at (0, 4).

For \( g(x) \):

• Slope \( a = 0.25 \), producing a gentler slope.
• Y-intercept \( b = -1 \), crossing the y-axis at (0, -1).

When comparing these linear functions, look at how their slopes and intercepts affect the graph's geometric properties, particularly when determining inverse relationships.

Reflection Across Line y=x

Reflection across the line \( y = x \) is a crucial test to check whether two functions are inverses.
This concept reveals if one function can be transformed into the other by reflecting over the diagonal line \( y = x \). To determine if \( f \) and \( g \) are inverses:

• Plot both functions on the same graph
• Observe if \( f(x) \) and \( g(x) \) are mirror images across the line \( y = x \)

If the graphs of \( f \) and \( g \) show perfect mirror symmetry with the line \( y = x \) acting as the mirror, then the functions are inverses.
If they do not reflect symmetrically, they are not inverses.
This method serves as a visual confirmation and breakdown of mathematical inverse relationships, aiding in deeper understanding and verification through graphical analysis.