Problem 61 - College Algebra

Question

Use a graphing utility to graph \(f\) and \(g\) in 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

Yes, the functions \(f(x) = 4x + 4\) and \(g(x) = 0.25x - 1\) are inverses of each other as confirmed by graphing them and the line \(y=x\) and determining that they reflect across the line \(y=x\).

Step 1: Recognize the functions

Recognize that \(f(x) = 4x + 4\) and \(g(x) = 0.25x - 1\) are linear functions. Linear functions have a straight line graph.

Step 2: Graph the functions

Use a graphing utility to plot both \(f(x) = 4x + 4\) and \(g(x) = 0.25x - 1\). Note how the graphs look.

Step 3: Draw the line y=x

In the same graphing utility, plot the line \(y=x\). This is the line of reflection that the two function graphs need to be mirrored across to be inverses.

Step 4: Determine if f and g are inverses

Analyze the graphs. If the graph of g is a reflection of f across the line y=x, then f and g are inverses. Look to see if for every x, where the function f(x) has a value of y, the function g(x) has a value of y for the corresponding x on the other side of the line y=x. The negative reciprocal of the slope of the linear function \(f(x)\) should be the slope of \(g(x)\). So if \(f(x)\) has a slope of 4, then \(g(x)\) should have a slope of 0.25, which is the case here confirming visually that these functions are indeed inverses.

Key Concepts

Linear Functions

Linear functions are equations that create a straight line when graphed on the coordinate plane. They have the general form \(y = mx + c\), where \(m\) represents the slope of the line and \(c\) is the y-intercept.
For example, in the function \(f(x) = 4x + 4\), 4 is the slope, and the line crosses the y-axis at 4, which is the y-intercept. \(g(x) = 0.25x - 1\) is another linear function where the slope is 0.25, and the y-intercept is -1.

• Linear functions always produce graphs that are straight lines.
• The steepness of the line is determined by the slope (\(m\)).
• The y-intercept (\(c\)) is where the line crosses the y-axis.

Understanding the components of linear functions is critical as it helps in plotting them accurately and analyzing their behavior on the graph.

Reflection Across y=x

The line \(y=x\) is significant because it is the line over which two inverse functions reflect. To check if two functions are inverses, you can graph both functions and the line \(y=x\).
When graphed, if \(f(x)\) and \(g(x)\) are inverses, the graph of \(g(x)\) will look like a mirror image of \(f(x)\) across the line \(y=x\). This visual cue is crucial in identifying inverse relationships.

• If every point \((a,b)\) on \(f(x)\) has a corresponding point \((b,a)\) on \(g(x)\), they are likely inverses.
• The line \(y=x\) serves as a symmetry axis for this reflection process.
• This reflection property holds true for all inverse functions and is especially easy to observe with linear functions.

Graphing Utility

A graphing utility is a tool that allows you to visualize mathematical functions and their relationships. It can be an online software, a physical calculator, or a computer program.
For functions like \(f(x) = 4x + 4\) and \(g(x) = 0.25x - 1\), graphing utilities help plot these equations to visually explore the relationship between them and any potential line of reflection like \(y=x\).

• Graphing utilities provide a detailed visual representation of functions.
• They allow for easy comparison between multiple graphs in a singular view.
• These tools can also help test different values, verify algebraic calculations, and analyze the behavior of functions.

Using a graphing utility is a practical way to interpret and analyze complex mathematical problems.

Slope Analysis

Slope is a crucial aspect of linear functions, representing the rate at which a line rises or falls as you move along it. For any line, the slope (\(m\)) is calculated by the change in \(y\) over the change in \(x\) (rise over run).
In our context, \(f(x) = 4x + 4\) has a slope of 4, and \(g(x) = 0.25x - 1\) has a slope of 0.25. Notice that one is the negative reciprocal of the other, \(4 = \frac{1}{0.25}\), confirming their inverse nature.

• The slope tells us the steepness and direction of a line's inclination.
• For inverse linear functions, the slopes are negative reciprocals.
• A steep positive slope indicates a steeper incline, while flatter slopes indicate gentler inclines.

Understanding slopes helps in predicting how two functions might relate to each other, such as being inverses if their slopes are negative reciprocals.