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.