Which of the following graphs represents a function 𝑓 of 𝑥 and its inverse function the inverse 𝑓 of 𝑥?
Let’s begin by reminding ourselves what it means for us to be talking about the inverse function. Suppose we have some function 𝑓. The inverse function, represented by the superscript negative one, undoes that original function. So the inverse function of 𝑓 of 𝑥 is equal to 𝑥 for all values of 𝑥 in the domain of the function. But what does this actually mean if we think about the graphical representation of these functions?
Well, if we plot 𝑦 equals 𝑓 of 𝑥 on the coordinate plane, we can find the graph of the inverse function 𝑦 is equal to the inverse 𝑓 of 𝑥 by reflecting the original graph across the line 𝑦 equals 𝑥. So let’s draw the line 𝑦 equals 𝑥 on each of our graphs and see if we can identify which one represents 𝑓 of 𝑥 and its inverse function. The line 𝑦 equals 𝑥 passes through the origin, the point zero, zero, and it has a slope of one, as shown.
Now, in fact, we can see that only one of these graphs represents a pair of functions which have been reflected across the line 𝑦 equals 𝑥. Specifically, we see that this is graph one. We can check this by identifying a couple of points that lie on each of these graphs. For instance, it appears as if the point negative 7.50 lies on the blue plot in graph one. Reflecting this point across the line 𝑦 equals 𝑥 essentially has the result of switching the 𝑥- and 𝑦-values. And we do indeed see that we get the point with coordinates zero, negative 7.5. So the graph that represents a function 𝑓 of 𝑥 and its inverse function is graph one.
Now, we note it doesn’t actually matter which of these plots represents the inverse function and which of the plots represents the original function. The function and its inverse function will always be inverses of one another.