We want to determine the composite function values using the provided table. Notice our
composite functions are written using this notation here on the left, which
we can rewrite as the notation given here on the right. Let's go ahead and do that to begin with.
So, we can write F of G of -2 using this notation here. G of F of 3 can be written using this
notation here. F of F of 0 can be written like this. And G of G of -4 could be written like this.
When evaluating a composite function, it's important to recognize which function to evaluate
first. When it's written using this notation here, it's probably easier to recognize that
we'll always start with the inner function, or in this case, G of -2. Another way to remember this
is to read this as F after G of -2, whether it's in this form here or this form here.
So, to evaluate this composite function, we'll first determine G of -2. We go over to our table,
find an input or x value of -2, which is here, and then find the output for function G,
which is positive 1. So, we can replace G of -2 with positive 1. This is equal to F of 1.
Now, we'll go back to our table and find the x value of positive 1,
and then find the output for function F. So, here's x value 1, and for function F,
the output is positive 3. So, this is equal to 3, which means F of G of -2,
or our composite function, is equal to positive 3 when the initial input is x = -2.
Now we have G of F of 3. So, we'll first determine the value of F of 3. So,
we'll find an x value of positive 3, which is here, then the output
for function F is positive 4. So, we'll replace F of 3 with 4, giving us G of 4.
To evaluate G of 4, we find the input or x value of positive 4, which is here,
then for function G, the output is positive 2, which is the value of the composite function
when x equals 3. So, we can write this as G of F of 3 is equal to positive 2.
Next, we have F of F of 0. So, we'll first determine the value of F of 0. So,
we'll find the output for function F when x equals 0. Well, here's x value 0;
here's function F, so the output is positive 1.
So, we substitute 1 for F of 0, giving us F of 1, which we already found earlier. When x is equal
to 1, the output for function F is positive 3. So again, we can say F of F of 0 is equal to 3.
And then for our last example, G of G of -4. We'll first determine G of -4. When x is -4,
the output for function G is positive 3. So, this would be equal to G of 3.
And then when x is positive 3, the output of function G is 0.
So, we can say G of G of -4 is equal to 0.
I hope you found this helpful.