Overview
This lesson explains composite functions, how to evaluate them, and demonstrates the process with worked examples.
Composite Functions Basics
- A composite function is written as f(g(x)), meaning g(x) is input into f(x).
- The symbol for composition is an open circle: f ∘ g or f(g(x)), not multiplication.
- To compute f(g(x)), substitute g(x) into every x in f(x).
Example: Calculating f(g(x)) and g(f(x))
- For f(x) = 3x – 4 and g(x) = x² – 3:
- f(g(x)) = 3(x² – 3) – 4 = 3x² – 9 – 4 = 3x² – 13.
- g(f(x)) = (3x – 4)² – 3.
- Expanding (3x – 4)² gives 9x² – 24x + 16; subtract 3 to get 9x² – 24x + 13.
Evaluating Composite Functions at Specific Values
- For f(x) = 5x + 2 and g(x) = x³ – 4, find f(g(2)):
- First, g(2) = 2³ – 4 = 8 – 4 = 4.
- Next, f(4) = 5×4 + 2 = 20 + 2 = 22.
- To find g(f(–1)):
- First, f(–1) = 5×(–1) + 2 = –5 + 2 = –3.
- Next, g(–3) = (–3)³ – 4 = –27 – 4 = –31.
Key Terms & Definitions
- Composite function — A function formed by applying one function to the result of another: f(g(x)).
- Function notation — Symbols such as f(x) and g(x) describe rules for input-output relationships.
Action Items / Next Steps
- Practice evaluating composite functions with new functions and input values.
- Review function notation and how to substitute one function into another.