Lesson 20: Composition of Functions
A composition of functions combines two functions so that the output of one function becomes the input of another.
It is written as (f ∘ g)(x) and read "f composed with g of x." Composition is an important tool for modeling complex relationships.
Definition
If you have functions f(x) and g(x), then the composition is:
(f ∘ g)(x) = f(g(x))
- First, apply
g(x)to x. - Then, apply
fto the result ofg(x).
Examples
Example 1:
Let f(x) = 2x + 3 and g(x) = x². Then:
(f ∘ g)(x) = f(g(x)) = f(x²) = 2x² + 3
(g ∘ f)(x) = g(f(x)) = g(2x + 3) = (2x + 3)²
Example 2:
Let f(x) = √x and g(x) = x + 5. Then:
(f ∘ g)(x) = √(x + 5)
(g ∘ f)(x) = √x + 5
Domain Considerations
The domain of (f ∘ g)(x) is all x-values in the domain of g such that g(x) is in the domain of f.
Example: f(x) = √x, g(x) = x − 3.
Then (f ∘ g)(x) = √(x − 3) → domain: x − 3 ≥ 0 → x ≥ 3.
Practice Problems
- f(x) = 3x − 1, g(x) = x². Find (f ∘ g)(2) and (g ∘ f)(2).
- f(x) = x + 4, g(x) = √x. Find (f ∘ g)(9) and its domain.
- f(x) = 1/x, g(x) = x − 2. Find (f ∘ g)(x) and determine its domain.
- f(x) = x², g(x) = 2x + 1. Find (g ∘ f)(x).