What is a composite function?

A composite function is created when you combine two functions by plugging one function into another function.

• Think of it as a “function machine inside another function machine.”

• Notation:

$$(f\circ g)(x)=f(g(x))$$

• g(x) goes first, then the output of g(x) is used as input for f(x).

Step 1: Understand the notation

• Read as: “f composed with g of x”

• Means: put g(x) into f(x)

Example:

$$(f\circ g)(x)=f(g(x))=f(x-1)$$

• Substitute x - 1 into f(x):

$$f(x-1)=2(x-1)+3=2x-2+3=2x+1$$

✅ So (f ∘ g)(x) = 2x + 1

Step 2: Understand order matters

• (f ∘ g)(x) ≠ (g ∘ f)(x) in most cases.

Example:

1. (f ∘ g)(x) = f(g(x)) = 2(x - 1) + 3 = 2x + 1

2. (g ∘ f)(x) = g(f(x)) = (2x + 3) - 1 = 2x + 2

• The results are different, so order is important!

Step 3: How to evaluate a composite function

Example:

• Find (f ∘ g)(2)

Step 1: Compute g(2)

$$g(2)=3(2)+1=7$$

Step 2: Plug g(2) into f

f(g(2))=f(7)=72=49f(g(2)) = f(7) = 7^2 = 49f(g(2))=f(7)=72=49

✅ So (f ∘ g)(2) = 49

Step 4: Composite functions with expressions

• You can compose functions with variables as well as numbers

Example:

• Find (f ∘ g)(x)

(f∘g)(x)=f(g(x))=f(x2)=2(x2)+5=2×2+5(f \circ g)(x) = f(g(x)) = f(x^2) = 2(x^2) + 5 = 2x^2 + 5(f∘g)(x)=f(g(x))=f(x2)=2(x2)+5=2x2+5

• Find (g ∘ f)(x)

(g∘f)(x)=g(f(x))=g(2x+5)=(2x+5)2(g \circ f)(x) = g(f(x)) = g(2x + 5) = (2x + 5)^2(g∘f)(x)=g(f(x))=g(2x+5)=(2x+5)2

✅ Notice (f ∘ g)(x) ≠ (g ∘ f)(x)

Step 5: Domain of composite functions

• The domain of (f ∘ g)(x) includes all x-values such that:

  1. g(x) is defined

  2. f(g(x)) is defined

Example:

• (f ∘ g)(x) = f(g(x)) = √(x - 3)

• Domain: x – 3 ≥ 0 → x ≥ 3

Why composite functions matter

• Used to model real-world situations with multiple steps

• Foundation for advanced algebra, calculus, and function transformations

• Useful for chaining operations or formulas

Real-life example:

• g(x) = temperature in °C given the day of the year x

• f(x) = electricity bill based on temperature

• (f ∘ g)(x) = electricity bill for day x

Common beginner mistakes

1. ❌ Forgetting the order of composition

2. ❌ Substituting incorrectly

3. ❌ Ignoring domain restrictions

4. ❌ Confusing (f ∘ g)(x) with f(x) + g(x)