Question Video: Evaluating Composite Functions at a Given Value Mathematics

Video Transcript

In the given figure, the red graph represents 𝑦 equals 𝑓 of 𝑥, while the blue represents 𝑦 equals 𝑔 of 𝑥. What is 𝑓 of 𝑔 of two?

We recall, first of all, that the notation 𝑓 of 𝑔 of two means we take the 𝑥-value two, apply the function 𝑔, first of all, and then apply the function 𝑓 to the result. Let’s begin then by finding the value of 𝑔 of two. Remember, 𝑔 is represented by the blue graph. So, we find two on the 𝑥-axis, move up to the blue graph ⁠— that’s this point here ⁠— and we see that the 𝑦-value, which will be 𝑔 of two, is one. So, by applying 𝑔 of 𝑥 to the value two, we obtain the value one. We’re now going to apply the function 𝑓 to the result. 𝑓 of 𝑔 of two becomes 𝑓 of one. We’ve replaced 𝑔 of two with the value we found of one.

We then return to the figure, and this time we’re looking at the red graph. We find one on the 𝑥-axis, move up to the red graph — that’s this point here — and we see that the 𝑦-value, representing 𝑓 of one, will be three. So, we find that 𝑔 of two is one, and then 𝑓 of 𝑔 of two, which is 𝑓 of one, is three. So, we’ve completed the problem. Now, just to demonstrate an important point, suppose we’ve done this in the opposite order. Suppose we thought that 𝑓 of 𝑔 of two meant we apply the function 𝑓 first.

Well, looking at two on the 𝑥-axis and going up to the red graph for 𝑓, we see that 𝑓 of two is equal to four. If we then applied the function 𝑔 to the result, which, in the correct notation, would be 𝑔 of 𝑓 of two, so that’s 𝑔 of four. We find four on the 𝑥-axis, go up to the blue graph, and we see that the 𝑦-value here is four. 𝑔 of 𝑓 of two then is four, whereas 𝑓 of 𝑔 of two is three. And so, we see that the order is extremely important when we’re composing functions. The correct answer, 𝑓 of 𝑔 of two, is three.