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Walk through examples, explana ons, and prac ce problems to learn how to find and evaluate composite func ons.
Given two func ons, we can combine them in such a way so that the outputs of one func on become the inputs of the other. This ac on defines a composite func on. Let's take a look at what this means!
Evalua ng composite func ons
Example
If and , then what is ?
Solu on
One way to evaluate is to work from the "inside out". In other words, let's evaluate first and then subs tute that result into to find our answer.
Let's evaluate.
Composing func ons
f (x) = 3x − 1 g(x) = x 3 + 2 f (g(3))
f (g(3)) g(3) f
g(3)
Since , then.
Now let's evaluate.
It follows that.
g(x)
g(3)
= x 3 + 2
= (3) 3 + 2 Plug in x =
g(3) = 29 f (g(3)) = f (29)
f (29)
f (x)
f (29)
= 3x − 1
= 3(29) − 1 Plug in x =
f (g(3)) = f (29) = 86
(^3) g 29 f 86
Since , we can subs tute in for .
This new func on should take directly to. Let's verify this.
Excellent!
Let's prac ce
Problem 1
g(x) = x 3 + 2 x 3 + 2 g(x)
f (g(x)) = 3(g(x)) − 1
= 3(x 3 + 2) − 1
= 3x 3 + 6 − 1
= 3x 3 + 5
f (g(x))
f (g(3))
= 3x 3 + 5
f (x) = 3x − 1
Problem 2
Composite func ons: a formal defini on
In the above example, we found and evaluated a composite func on.
g(x) = x 3 + 2
Evaluate g(f (1)).
Check
[I need help. Please show me the solu on.]
m(x) = 3x − 2
n(x) = x + 4
Find m(n(x)).
Check
[I need help. Please show me the solu on.]
We can find as follows:
Since we now have func on , we can simply subs tute in for to find.
Of course, we could have also found by evalua ng. This is shown below:
(h ∘ g)(x)
(h ∘ g)(x) = h(g(x))
= (g(x)) 2 − 2(g(x))
= (x + 4) 2 − 2(x + 4)
= x 2 + 8x + 16 − 2x − 8
= x 2 + 6x + 8
h ∘ g −2 x (h ∘ g)(−2)
(h ∘ g)(x)
(h ∘ g)(−2)
= x 2 + 6x + 8
(h ∘ g)(−2) h(g(−2))
The diagram below shows how is related to.
Here we can see that func on takes to and then func on takes to , while func on takes directly to.
Now let's prac ce some problems
Problem 3
(h ∘ g)(−2) = h(g(−2))
= h(2) Since g(−2) = −
= 0 Since h(2) = 2^2
(h ∘ g)(−2) h(g(−2))
−2 (^) g 2 h 0
h ∘ g
g −2 2 h 2 0 h ∘ g −2 0
f (x) = 3x − 5
g(x) = 3 − 2x
Challenge Problem
The graphs of the equa ons and are shown in the grid below.
Which of the following best approximates the value of?
Choose 1 answer:
y = f (x) y = g(x)
2
4
6
8
y
x
y = f (x)
y = g(x)
(f ∘ g)(8)
Check
A −
B −
C 0
D 2
