Problem 85 - 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. $$f(x)=\frac{1}{x}+2, g(x)=\frac{1}{x-2}$$

Step-by-Step Solution

Answer

Visually comparing the graphs of \(f(x)\) and \(g(x)\), after also plotting the line \(y=x\), should help determine whether or not \(f\) and \(g\) are inverses. This can be confirmed by checking if the points on the graphs of \(f(x)\) and \(g(x)\) are reflected about the line \(y=x\).

Step 1: Plot the Function \(f(x)\)

Plot the graph of \(f(x) = \frac{1}{x}+2\). A graphing utility should be used in this case, although a manual plot may also be generated by selecting various values of \(x\) and determining the corresponding \(f(x)\) values.

Step 2: Plot the Function \(g(x)\)

Likewise, plot the graph of \(g(x) = \frac{1}{x-2}\). Again, a graphing utility is advised to get a precise plot.

Step 3: Plot the line \(y=x\)

Plot the line \(y=x\). Every point \((a,a)\) lies on this line, it intercepts the origin (0,0), and has a slope of 1.

Step 4: Compare the graphs

Compare the three graphs. The main objective is to determine whether \(f\) and \(g\) are inverse functions. This can be verified visually through the test mentioned earlier - that is, if a point \((a,b)\) is on the graph of \(f\), then its inverse point \((b,a)\) should be on the graph of \(g\). Similarly, if a point \((p,q)\) is on the graph of \(g\), then its inverse point \((q,p)\) should be on the graph of \(f\). This means that the graph of the inverse function should be the reflection of the original function over the line \(y=x\).

Key Concepts

Graphing Utilities

Graphing utilities are powerful tools that help visualize mathematical behavior. These can be in the form of software applications or graphing calculators. They provide a dynamic way to plot complex functions quickly and accurately. No long calculations are needed - just plug in your function, and the graphing utility does the rest!
Here are some advantages of using graphing utilities:

• Speed: They rapidly produce graphs, saving you valuable time.
• Accuracy: These tools provide precise graph plots, minimizing errors that might arise in manual plotting.
• Education: Graphing utilities help in understanding complex function relationships by visualizing them.

In this exercise, using a graphing utility helps create the plots for both \(f(x) = \frac{1}{x} + 2\) and \(g(x) = \frac{1}{x-2}\), as well as the line \(y=x\). By overlaying these graphs, we can easily compare and analyze the functions to determine the nature of their relationship.

Function Graphing

Function graphing is the process of plotting the points of a function onto a coordinate plane. Understanding a function's graph is vital for interpreting its behavior across its domain. The function \(f(x) = \frac{1}{x} + 2\) creates a hyperbola shifted up by 2 units. This shift affects its asymptotes and the function's behavior visually.
Similarly, plotting \(g(x) = \frac{1}{x-2}\) reveals another hyperbola, but this one shifts 2 units right. This shift occurs because the formula manipulates the variable \(x\) inside \(\frac{1}{x-2}\).
It's essential to:

• Identify asymptotes: Both functions have asymptotes because they are rational functions with \(x\) in the denominator.
• Determine visual modifications: Note how transformations such as shifts or stretches affect the graphs.
• Compare functions: Looking at these plots can help spot inverse relationships.

Graphing both functions helps us to conceptualize their differences and similarities, which are crucial in understanding inverse functions.

Reflection over Line \(y=x\)

The line \(y=x\) is the defining line when determining if two functions are inverses. This line illustrates the perfect symmetry for such reflections. When graphing, the reflection property states:If a function \(f\) has a point \((a, b)\), its inverse \(g\) must have the point \((b, a)\). Conversely, if \(g\) has \((p, q)\), \(f\) should have \((q, p)\).
Here’s how you verify inverses using reflection:

• Plot both functions on the same graph as \(y=x\).
• Check if their graphs are symmetrical about the line \(y=x\).
• Inspect points on the graphs: For \(f(x)\) and \(g(x)\), find points which mirror each other over \(y=x\).

For example, if plotting \(f(x) = \frac{1}{x} + 2\) and \(g(x) = \frac{1}{x-2}\), a graphing utility might show they aren’t perfect mirrors over \(y=x\), indicating the absence of a strict inverse relationship. Understanding this concept helps verify functions' inverseness through visual confirmation.