8.2 Composite of Functions
Chapter 9 – Combinations of Functions
MHF 4U1: Advanced Functions
You have learned four ways of combining functions: sums, differences, products, and quotients.
Another type of combined function is a composite function.
A composite function is a function that depends on another function. It is formed when one function is
substituted into another.
Given two functions f(x) and g(x), the composite function is represented using the following notations:
(𝑓 ◦ 𝑔)(𝑥) OR 𝑓(𝑔(𝑥))
Read “f of g of x” OR
“the composition of f and g”
Input-output diagram
Note: the order of the functions is important. As read from left to right, the second (inner) function is
substituted into the first (outer) function.
f(g(x)) ≠ g(f(x)) The composition of these two functions generate different answers.
2
Example 1: Given 𝑓(𝑥) = 3𝑥 − 5, 𝑔(𝑥) = 𝑥 − 𝑥, and ℎ(𝑥) =
a) f(g(-1))
b) g(f(2))
𝑥+1
, find the following:
𝑥
c) (h◦f )(2)
d) Domain of h(f(x))
2
Example 2: Let 𝑓(𝑥) = 𝑥 − 2 and 𝑔(𝑥) = 𝑥 + 1. Determine an equation for each composite function, graph
the function and give its domain and range.
a)
(f◦f)(x)
b) g(f(x))
Domain:
Domain:
Range:
Range:
Example 3: For a car travelling at a constant speed, the distance driven, d kilometres, is represented by
𝑑(𝑡) = 80𝑡, where t is the time in hours. The cost of gasoline, in dollars, for the drive is represented by
𝐶(𝑑) = 0. 09𝑑. Determine C(d(5)).
A composite function combining a function and its inverse
Example 4: Let 𝑓(𝑥) =− 2𝑥 + 1 and 𝑔(𝑥) = 𝑥 + 2 determine
−1
−1
(a) 𝑓 (𝑥)
(b) 𝑓 (𝑓(𝑥))
−1
(c) 𝑓(𝑓 (𝑥))
In General:
Even though the order of the function in the composition is reversed, the results are the same.
The composite function is _____________________.
−1
(d) (𝑔 ◦ 𝑔 )(2)
2
Example 5: Let 𝑓(𝑥) = 𝑥 + 2𝑥 − 4 and 𝑔(𝑥) =
1
. Show that 𝑔(𝑓(𝑥)) is undefined for 𝑥 = 1 and 𝑥 =− 3.
𝑥+1
Consolidate: Page 552 # 1, 2a, 4, 5ace, 6ace, 7, 13, 14