If 𝑓 of 𝑥 equals three to the power of 𝑥 and 𝑔 of 𝑥 equals 𝑥 minus two, what is 𝑓 of 𝑔 of 𝑥?
Let’s just begin by recapping this notation, which are read as 𝑓 of 𝑔 of 𝑥. This means that we take an input value 𝑥, we apply the function 𝑔 to it, and then we apply the function 𝑓 to the result. This is called a composite function. Let’s see what that would look like then for these two functions 𝑓 and 𝑔.
We take an input value 𝑥 and apply the function 𝑔. Well, 𝑔 is the function which subtracts two from its input value. So, 𝑔 of 𝑥 is equal to 𝑥 minus two. Then, we need to apply the function 𝑓, by taking this expression for 𝑔 of 𝑥 — that’s 𝑥 minus two — as the input. So, 𝑓 of 𝑔 of 𝑥 is the same as 𝑓 of 𝑥 minus two. Now 𝑓 of 𝑥 is the function three to the power of 𝑥. It’s three to the power of whatever the input value is. So, 𝑓 of 𝑥 minus two is three to the power of 𝑥 minus two.
So, the composite function 𝑓 of 𝑔 of 𝑥, which is the function obtained when we take an input value, apply the function 𝑔, and then apply the function 𝑓 to the result, is three to the power of 𝑥 minus two.