Creating New Functions by Combining Others

What is Function Composition?

Definition: The composition of two functions f and g creates a new function where the output of g becomes the input of f.

Notation:

(f∘g)(x)=f(g(x))

The Mechanics of Composition

A. Basic Composition Process

To find (f∘g)(x):

1. Start with the inner function g(x)

2. Substitute g(x) into f(x) wherever you see x

3. Simplify the resulting expression

Example 1: Let $$f(x)=x^2+1$$ and $$g(x)=2x-3$$

(f∘g)(x)=f(g(x))=f(2x−3)=$$(2x-3)^2+1=4x^2-12x+10$$

Domain of Composite Functions

Critical Concept: The domain of (f∘g)(x) must satisfy BOTH:

1. x is in the domain of $g$

2. g(x) is in the domain of $f$

Step-by-step domain analysis:

1. Find domain of g(x)

2. Find domain of f(x)

3. Restrict domain of g so that g(x) is in domain of f

Example 3: $$f(x)=\sqrt{x} , g(x)=x-4$$

• Domain of $g$: all real numbers

• Domain of f: x≥0

• For (f∘g)(x)=$$\sqrt{x-4}$$​, need x−4≥0⇒ x≥4

• Domain: [4,∞)

Evaluating Composite Functions at Specific Values

Two Methods:

1. Find formula first, then substitute

2. Chain evaluation: Start inside, work outward

Example 4: If $$f(x)=x^2$$ and $$g(x)=x+2$$, find (f∘g)(3)

Method 1 (formula):
(f∘g)(x)=$$(x+2)^2 \rightarrow (3+2)^2=25$$

Method 2 (chain):
g(3)=3+2=5 f(5)=25

Multiple Compositions

We can compose three or more functions:

(f∘g∘h)(x)=f(g(h(x)))

Work from inside to outside

Example 5: $$f(x)=\sqrt{x} , g(x)=x+1 , h(x)=2x$$

(f∘g∘h)(x)=f(g(h(x)))=f(g(2x))=f(2x+1)=$$\sqrt{2x+1}$$​

Decomposition: Breaking Apart Composite Functions

Often we need to decompose a function into simpler parts.

Example 6: Decompose H(x)=$$\sqrt{3x^2+1}$$​
Possible decomposition:
g(x)=$$3x^2+1$$ (inner)
f(x)=$$\sqrt{x}$$ (outer)
Then H(x)=(f∘g)(x)

Special Composition Results

Identity Function

If I(x)=x, then:

(f∘I)(x)=(I∘f)(x)=f(x)

Inverse Functions

If f and g are inverses, then:

(f∘g)(x)=(g∘f)(x)=x

Test for inverses: Check if f(g(x))=x and g(f(x))=x

VIII. Graphical Interpretation

To understand $(f\circ g)(a)$ graphically:

1. Start at $x=a$ on x-axis

2. Find $g(a)$ on graph of $g$ (y-value)

3. Use that y-value as input to $f$

4. Read final output from graph of $f$

This is why composition represents a chain of processes!

Common AP Exam Questions

Type 1: Find the Composition

Given algebraic expressions for $f$ and $g$, find (f∘g)(x) or (g∘f)(x)

Type 2: Evaluate at a Point

Given tables, graphs, or equations, find value of composition at specific input

Type 3: Domain Analysis

“Find the domain of (f∘g)(x)“

Type 4: Decomposition

“Express H(x)=$$\frac{1}{\sqrt{x^2+4}}$$​ as composition of two functions”

Type 5: Application Problems

Word problems involving chains of processes

Step-by-Step Problem Solving Strategy

For composition problems:

1. Identify inner and outer functions

2. Substitute carefully (use parentheses!)

3. Simplify the resulting expression

4. Check domain restrictions from both functions

5. Verify with specific values if possible