### Video Transcript

In the given figure, the red graph
represents π¦ equals π of π₯, while the blue represents π¦ equals π of π₯. What is π of π of two?

We recall, first of all, that the
notation π of π of two means we take the π₯-value two, apply the function π,
first of all, and then apply the function π to the result. Letβs begin then by finding the
value of π of two. Remember, π is represented by the
blue graph. So, we find two on the π₯-axis,
move up to the blue graph β β thatβs this point here β β and we see that the π¦-value,
which will be π of two, is one. So, by applying π of π₯ to the
value two, we obtain the value one. Weβre now going to apply the
function π to the result. π of π of two becomes π of
one. Weβve replaced π of two with the
value we found of one.

We then return to the figure, and
this time weβre looking at the red graph. We find one on the π₯-axis, move up
to the red graph β thatβs this point here β and we see that the π¦-value,
representing π of one, will be three. So, we find that π of two is one,
and then π of π of two, which is π of one, is three. So, weβve completed the
problem. Now, just to demonstrate an
important point, suppose weβve done this in the opposite order. Suppose we thought that π of π of
two meant we apply the function π first.

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