In the given figure, the green points represent the function 𝑓 of 𝑥. Do the blue points represent the inverse 𝑓 of 𝑥?
This notation 𝑓 with a superscript negative one is the inverse 𝑓 of 𝑥. And we recall that the inverse function essentially undoes the original function. But what does this mean graphically? There are two ways to think about it. If we plot the graph of the function 𝑦 equals 𝑓 of 𝑥, then we can also draw the graph of 𝑦 is equal to the inverse 𝑓 of 𝑥 by mapping a reflection across the line 𝑦 equals 𝑥. Let’s draw lines through our points and identify whether this is happening.
It certainly does appear as if the green line can be reflected across the line 𝑦 equals 𝑥 onto the blue line. But we can check by considering individual points. Take, for instance, the point eight, four that lies on the line 𝑦 equals 𝑓 of 𝑥. If this point is reflected across the line 𝑦 equals 𝑥, that results in us switching the 𝑥- and 𝑦-values. In other words, reflecting the point eight, four across the line 𝑦 equals 𝑥 gives us the point four, eight. That does indeed lie on the blue line. Similarly, consider the point four, two. If we reflect this point across the line 𝑦 equals 𝑥, we reverse 𝑥 and 𝑦. And this gives us the point with coordinates two, four.
So, in fact, we can see that if we reflect the line 𝑦 equals 𝑓 of 𝑥 across the line 𝑦 equals 𝑥, we do get the blue line. This means the blue line must be 𝑦 is equal to the inverse 𝑓 of 𝑥. And so the answer is yes. The blue points represent the inverse 𝑓 of 𝑥.