Problem 71 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the inverse function of \(f\) algebraically. Use a graphing utility to graph both \(f\) and \(f^{-1}\) in the same viewing window. Describe the relationship between the graphs. $$f(x)=4 x-9$$

Step-by-Step Solution

Answer

The inverse function of \(f(x)=4x-9\) is \(f^{-1}(x)=(x+9)/4\). When graphing \(f\) and \(f^{-1}\), the two functions reflect each other about the line \(y=x\).

Step 1: Find the inverse function algebraically

The function to be inversed is given as \( f(x)=4x-9 \). To find the inverse, first, replace \(f(x)\) with \(y\), so the equation becomes \( y=4x-9 \). Then swap \(x\) and \(y\) in the equation, resulting in \( x=4y-9 \). Solve this equation for \(y\) to obtain the inverse function. With a few algebraic manipulations, \(y\) can be isolated as follows: \( 4y=x+9 \). Dividing through by 4 results in \( y = (x+9)/4 \). Thus, the inverse function \(f^{-1}\) is \( f^{-1}(x)=(x+9)/4 \).

Step 2: Graph both functions

Utilize a graphing utility to graph the original function \(f(x)\) and its inverse \(f^{-1}(x)\) in the same viewing window. The line of the original function will be straight and slanted, as it is a linear function while the line for the inverse function will also be straight and slanted but at a different angle.

Step 3: Describe the relationship between the two graphs

The plot of \(f\) and \(f^{-1}\) will reveal that they reflect each other over the line \(y=x\). This is a characteristic of functions and their inverse; they are always mirror images of each other with respect to the line \(y=x\).

Key Concepts

Linear Functions

A linear function is one of the simplest types of functions. It can be written in the form of \( f(x) = ax + b \). Here, \( a \) and \( b \) are constants, where \( a \) is the slope, and \( b \) is the y-intercept. This type of function graphs as a straight line. In our exercise, the original function is \( f(x) = 4x - 9 \). The slope is 4, indicating how steep the line is, and the y-intercept is -9. This tells us that when \( x \) is 0, the function crosses the y-axis at -9. Linear functions are easy to work with because their graphs are straightforward lines. They're predictable and show a constant rate of change. This makes it much simpler to find their inverse, as the inverse will also be a linear function. The slope of the inversed function \( f^{-1}(x) \) is \( \frac{1}{a} \), leading to different angles but still maintaining a straight line characteristic. Understanding linear functions is the first step to grasping their inverses.

Graphing Utility

Graphing utilities are powerful tools that help visualize mathematical functions, such as linear functions and their inverses. These utilities can take the form of physical graphing calculators or software tools available on computers or mobile devices. Using a graphing utility, you can easily plot the original function and its inverse to observe their characteristics. For the function \( f(x) = 4x - 9 \) and its inverse \( f^{-1}(x) = \frac{x+9}{4} \), the graphing utility will show two straight lines in a shared window. Such visuals provide immediate insight into the relationship between the two functions. You can see how one line passes through certain points and compare angles and intersections with the axes. This can help you better understand the concept of inverse functions and their properties. Graphing utilities make it easy to see mathematical relationships "live," thus deepening your comprehension of these functions.

Reflection over y=x

In mathematics, every function and its inverse have a unique geometric relationship: they reflect over the line \( y = x \). This means if you were to draw the line \( y = x \) on a graph, the graph of the function and its inverse would be mirror images of each other.For our exercise, we consider the function \( f(x) = 4x - 9 \) and its inverse \( f^{-1}(x) = \frac{x+9}{4} \). When plotted, you will notice that these two functions reflect across the line \( y = x \). This property visually confirms that you've correctly calculated the inverse function.Reflection over \( y = x \) signifies that if you took a point \( (a, b) \) on the original function, the inverse function would pass through the point \( (b, a) \). Understanding this reflection helps ensure the integrity of your function manipulations. This characteristic is fundamental when working with inverses, as it provides a visual and mathematical check for accuracy.