Question

In Exercises 27-34, find the inverse function of \(f\). Graph (by hand) \(f\) and \(f^{-1}\). Describe the relationship between the graphs. $$ f(x)=2 x-3 $$

Short answer

The inverse function of \(f(x) = 2x - 3\) is \(f^{-1}(x) = (x + 3)/2\). The graph of these functions show that they are mirror images of each other along the line \(y = x\), a defining characteristic of inverse functions. They intersect at the point (1,1).

Step-by-step solution

Find the inverse

First, let's find the inverse function. You start by replacing \(f(x)\) with \(y\), hence \(y = 2x - 3\). Next, swap the positions of \(y\) and \(x\) to get \(x = 2y - 3\). Lastly, adjust to get \(y\) on one side of the equation. Thus, the inverse function, denoted as \(f^{-1}(x)\), is \(y = (x+3)/2\)

Graph the functions

To graph \(f(x)\) and \(f^{-1}(x)\), pick a range of x-values and find the corresponding y-values for both functions. A good range is from -3 to 3. For \(f(x)\), the line starts at \(y = -3\) and has a slope of 2, indicating it increases as \(x\) increases. For \(f^{-1}(x)\), the line starts at \(y = 1.5\) and has a slope of 0.5. The range chosen affects the details of the graph, but the general shape and slope remains the same.

Describe the relationship between the graphs

The graphs of \(f(x)\) and \(f^{-1}(x)\) are reflections of each other over the line \(y = x\). This is the defining characteristic of inverse functions - they're always mirror images over the \(y = x\) line. They also intersect at the point where \(x = y\). In this case, they intersect at (1,1).

Key concepts

Graphing Inverse Functions

Graphing inverse functions is a visual way to understand the relationship between a function and its inverse. To graph the original function, like f(x) = 2x - 3, you would typically plot a few key points and draw the line that represents the equation. To graph its inverse, f^-1(x) = (x + 3) / 2, you follow a similar process, making sure to switch the x's and y's.

For instance, if you select values for x in f(x), to find those same values in f^-1(x), you would treat them as y-values and solve for x. This will give you pairs of points that you can then plot on the same set of axes. What you'll notice is that each function is a straight line, and the inverse function will always have points that correspond to the original function with their x and y values reversed.

Properties of Inverse Functions

Defining Property of Inverse Functions

The main property is that applying a function followed by its inverse will return the original value. This means for a function f and its inverse f^-1, f(f^-1(x)) = x, and f^-1(f(x)) = x. In terms of the graph, this implies that if a point (a, b) lies on the graph of f, then the point (b, a) will lie on the graph of f^-1.

One-to-One Functions

Another important property to understand is that only one-to-one functions have inverses that are also functions. A one-to-one function is one where each x-value maps to a unique y-value. This ensures that the inverse will pass the vertical line test and thus be a valid function. The original exercise, f(x) = 2x - 3, is one-to-one, which is why its inverse is also a function.

Finding Inverse Functions

Finding the inverse of a function involves a few key steps. To find the inverse of the function given in the exercise, f(x) = 2x - 3, you begin by replacing f(x) with y, giving you y = 2x - 3. The next step is to swap x and y, resulting in x = 2y - 3. Finally, you solve this new equation for y, which gives you the inverse function f^-1(x) = (x + 3) / 2. It's important to then check your work by composing f(f^-1(x)) and f^-1(f(x)) to ensure they both simplify to x. Remember, in an inverse function, x becomes the output and y becomes the input. This is the critical switch that reverses the process of the original function.

Reflective Symmetry in Inverse Functions

Reflective symmetry is a fascinating aspect of inverse functions depicted on a graph. For any function and its inverse, they will have symmetry along the line y = x. In our example, the function f(x) = 2x - 3 and its inverse f^-1(x) = (x + 3) / 2 will reflect over this line. When graphing by hand, as in the provided solution steps, it is helpful to use this line as a reference to ensure accuracy.

This reflection creates a situation where if you draw the line y = x and then fold the graph along it, the function and its inverse would line up perfectly. This symmetry is not just a visual characteristic but also an essential hallmark of functions and their inverses. In contrast, functions that do not have this reflective symmetry with respect to the line y = x do not have inverse functions that are also functions.