Let 𝑓 and 𝑔 be differentiable functions such that the inverse of 𝑓 of 𝑥 is equal to 𝑔 of 𝑥 for all 𝑥. The given table shows selected values of 𝑓 of 𝑥 and 𝑓 prime of 𝑥. What is the value of 𝑔 prime of three? And then, we have a table here which contains values for 𝑥, 𝑓 of 𝑥, and 𝑓 prime of 𝑥.
Let’s begin by considering what we know about our functions 𝑓 and 𝑔. They’re differentiable functions. And we know that the inverse of 𝑓 is equal to this other function 𝑔. Geometrically, what does that mean? The inverse of a function can be interpreted as its reflection in the line 𝑦 equals 𝑥. So since 𝑔 and 𝑓 are inverses for one another, we can use the values in the table to find some values for 𝑔 of 𝑥.
We know that 𝑓 of three is equal to four. So we can plot that as the point with the Cartesian coordinates three, four. If we reflect this point in the line 𝑦 equals 𝑥, we get four, three. And since our function 𝑔 is a reflection of the function 𝑓 in the line 𝑦 equals 𝑥, we now know that 𝑔 of four must be equal to three. We know that 𝑓 of five is equal to three. And this is represented by the point with Cartesian coordinates five, three. Reflecting this point in the line 𝑦 equals 𝑥 and we end up with the point with Cartesian coordinates three, five. So that tells us that 𝑔 of three is equal to five.
Now, we’re trying to find the value of 𝑔 prime of 𝑥. So next, we recall something called the inverse function theorem. And this says that if 𝑓 prime of 𝑥 is not equal to zero, then the slope or the gradient of the inverse of 𝑓 at 𝑓 of 𝑥 is equal to one over 𝑓 prime of 𝑥. Now, we know that the inverse of 𝑓 is equal to 𝑔 in our question. So we can replace the inverse of 𝑓 with 𝑔 in the theorem. And this time, we see that the slope of 𝑔, and that’s remember 𝑔 prime, at 𝑓 of 𝑥 is equal to one over 𝑓 prime of 𝑥.
Well, 𝑓 of five corresponds to 𝑔 of three. So the slope of 𝑔 at 𝑓 of five is equal to one over 𝑓 prime of 𝑥, which we now know is one over 𝑓 prime of five. 𝑓 prime of five is negative four. So the slope of 𝑔 at 𝑓 of five, which we know is three, is equal to one over negative four. And that’s negative one-quarter.
So 𝑔 prime of three is negative one-quarter.