### Video Transcript

If π of π₯ equals three minus π₯
squared and π of π₯ equals two π₯ plus four, find π of π of one.

Now, letβs just be clear on the
notation used in the question. π of π of one means the composite
function π of π of π₯ evaluated at π₯ equals one. We may also see this written with a
small circle between the two letters. Now, what we need to remember is
that the composite function π of π of π₯ means the function we get when we apply
π first and then apply π to the result. It doesnβt mean the product of the
functions π and π. Weβre going to look at two ways of
answering this question. In the first method, weβre going to
substitute π₯ equals one right at the start. So, weβre going to find π of one
and then evaluate π for this value. π of π₯ is the function two π₯
plus four, so π of one will be two multiplied by one plus four, which is equal to
six.

Weβre now going to take this value
and evaluate the function π. So, π of π of one will become
simply π of six. π of π₯ is the function three
minus π₯ squared, so π of six will be three minus six squared. Thatβs three minus 36, which is
equal to negative 33. So, in this method, we evaluated π
of one, first of all, and then we took this as our input for the second
function. Overall, we found that π of π of
one is equal to negative 33. The second approach we could take
is to find a general algebraic expression for the composite function π of π of π₯
and then evaluate it when π₯ is equal to one. This is probably more complicated
in this case, but this method would be useful if we were asked to find π of π of
π₯ for multiple different π₯-values.

So, letβs see what this looks
like. Weβre looking to find the general
composite function π of π of π₯. Remember, π of π₯ is the function
two π₯ plus four. So, replacing π of π₯ with two π₯
plus four, weβre now looking to find the function π of two π₯ plus four. What we do then is we take the
expression two π₯ plus four as our input to the function π. π of π₯ is the function three
minus π₯ squared. So, π of two π₯ plus four is the
function three minus two π₯ plus four all squared. We can keep our composite function
in this form or we can distribute the parentheses and simplify, if we wish. And it will give π of π of π₯
equals negative four π₯ squared minus 16π₯ minus 13.

We now have a general expression
for π of π of π₯. But remember, we were asked to
evaluate this when π₯ equals one. So, the final step is to substitute
π₯ equals one. This gives negative four minus 16
minus 13, which is equal to negative 33, the same answer as we found using our
previous method. So in both cases, we found that π
of π of one is equal to negative 33. The first method is probably more
straightforward for this particular question. But if we needed to evaluate π of
π of π₯ for multiple π₯-values, then the second method may well be more
efficient.