What is a Composite Function?

A composite function is formed when the output of one function becomes the input of another. If you have functions f and g, the composite function f(g(x)) means "first apply g to x, then apply f to the result".

(f ∘ g)(x) = f(g(x))

Read as "f composed with g of x" or "f of g of x"

How Composition Works

Input

x

Apply g

g(x)

Apply f

f(g(x))

Output

y

Important: In general, f(g(x)) ≠ g(f(x)). The order of composition matters! Composition is not commutative.

Finding Composite Functions

To find f(g(x)), replace every x in f with the entire expression g(x):

Write out the formula for f(x).
Wherever you see x in f, substitute g(x) using brackets.
Simplify the resulting expression.

Example: If f(x) = x² + 1 and g(x) = 3x - 2, find f(g(x)).

f(g(x)) = f(3x - 2) = (3x - 2)² + 1

= 9x² - 12x + 4 + 1

= 9x² - 12x + 5

Domain of Composite Functions

The domain of f(g(x)) must satisfy two conditions:

Condition 1

x must be in the domain of g (so g(x) is defined).

Condition 2

g(x) must be in the domain of f (so f(g(x)) is defined).

For example, if f(x) = √x and g(x) = x - 3, then f(g(x)) = √(x - 3). We need x - 3 ≥ 0, so the domain is x ≥ 3 or [3, ∞).

Key Vocabulary

Composite Function

A function formed by applying one function to the output of another, written as f(g(x)) or (f ∘ g)(x).

Inner Function

The function that is applied first. In f(g(x)), g is the inner function.

Outer Function

The function that is applied second. In f(g(x)), f is the outer function.

Composition

The operation of chaining functions together so one's output feeds into the next's input.

Worked Examples

1

If f(x) = 2x + 3 and g(x) = x², find f(g(x)) and g(f(x)).

f(g(x)): Replace x in f with g(x) = x²: f(x²) = 2(x²) + 3 = 2x² + 3.

g(f(x)): Replace x in g with f(x) = 2x+3: g(2x+3) = (2x+3)² = 4x² + 12x + 9.

Note: f(g(x)) ≠ g(f(x)) — the order matters!

2

If f(x) = x + 4 and g(x) = 3x - 1, evaluate f(g(2)).

Step 1: First calculate g(2): g(2) = 3(2) - 1 = 5.

Step 2: Then calculate f(5): f(5) = 5 + 4 = 9.

Answer: f(g(2)) = 9.

3

Find the domain of f(g(x)) where f(x) = √x and g(x) = 5 - 2x.

Step 1: f(g(x)) = √(5 - 2x).

Step 2: Domain of g: all real numbers (no restriction).

Step 3: We need g(x) ≥ 0 for f: 5 - 2x ≥ 0, so x ≤ 5/2.

Answer: Domain = (-∞, 5/2] or {x : x ≤ 2.5}.

Knowledge Check

Select the correct answer for each question. Click "Check Answer" to see if you are right.

Question 1

If f(x) = x + 2 and g(x) = 3x, what is f(g(4))?

14 18 10 6

Question 2

If f(x) = x² and g(x) = x - 1, what is f(g(x))?

x² - 1 Both b and c x² - 2x + 1 (x - 1)²

Question 3

Is f(g(x)) always equal to g(f(x))?

Yes, always Only for linear functions No, generally not Only for quadratic functions

Question 4

If f(x) = √x and g(x) = x + 4, what is the domain of f(g(x))?

x ≥ 0 All real numbers x ≥ 4 x ≥ -4

Question 5

If f(x) = 2x - 1 and g(x) = x + 5, what is g(f(3))?

10 15 8 11

Composite Functions