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# Evaluating composite functions (advanced)

Video transcript

We are told that h
of x is equal to 3x. g of t is equal to negative
2t minus 2 minus h of t. f of n is equal to negative
5n squared plus h of n. So we have three
function definitions, and two of these
function definitions are actually defined in
terms of another function-- in particular, in terms
of the function h. And then we're asked to
calculate, what is h of g of 8? And this can be very
daunting, but we just have to remember that all a
function is is something that takes an input and gives
you an output for it. And this somewhat
convoluted looking statement is another way of
saying, look, we are going to take
the number 8, and we are going to input it
into the function g. And then that is going
to produce g of 8. And then we're going to
take whatever value that is and input that
into the function h. So we're going to
take this whole thing, and then we're going to input
that into the function h. And what we will have will
be h of what we inputted, h of g of 8. So let's just do it
one step at a time. Let's figure out what g of 8 is. And I'm going to color code it,
so we can keep track of things. g of 8 is equal to-- well, g of
t, we have our definition here. So our input now, 8 is going
to be our t, so our input is 8. So every place where we see a
t in this function definition, we replace it with an 8. So it's going to be negative
2 times 8 minus 2 minus-- and this might be
a little daunting, but let's just replace
this t with an 8 and then see if we
can make sense of it. h minus-- and let me do
it in the right color-- minus 2 minus h of 8. Notice, to evaluate g of 8,
all we did is, everywhere we saw a t, we replaced
it with the input 8. Now let's see if we
can calculate this. This is going to be equal
to negative 2 times 8 is negative 16. Minus 2 is negative 18. Let me do that the same. So this is going to be equals
negative 18 minus-- what is h of 8 going to be equal to? So let's do that over here. So h of 8-- this thing, h of 8. Now we go to the
definition of h. Don't worry about
later we're going to input all this
business into h again. Just let's worry about
it one step at a time. We need to calculate h of 8. So h of 8 is just going to be--
well, every time we see an x, we replace it with an 8--
it's going to be 3 times 8, which is equal to 24. So this value right
over here is 24. We are subtracting it,
so we have minus 24. Negative 18 minus 24 is what? That's negative 42. So all of this
business is going to be equal to-- did I do that right? Yeah-- negative 42. So we figured out
what g of 8 is. It is negative 42. So this right over
here is negative 42. And now we can input
negative 42 into h. Let me do it right over here.
h of negative 42-- remember, negative 42 is the
same thing as g of 8. So this is h of g of 8 is the
same thing as h of negative 42. Let me do that in
the same color. This is going to be equal to
3 times negative 42, which is equal to-- this
is negative 126. And we are done. So it seemed
convoluted at first, but if you just
keep track of what's our input, what's our
output, and really just evaluate the functions,
it should hopefully be reasonably straightforward.