Overview
This lecture introduces the concept of function composition, demonstrates how to evaluate composite functions, and walks through step-by-step examples.
Understanding Function Composition
- Composing functions means applying one function to the output of another, creating nested functions.
- The notation F(G(x)) means input x into G, then use G(x) as the input for F.
- Function composition can involve two or more functions.
Example: Evaluating Composite Functions
- To find F(G(2)), first compute G(2), then use that result as input for F.
- G(2) is -3; substitute into F: F(-3) = (-3)² - 1 = 9 - 1 = 8.
- Therefore, F(G(2)) = 8.
Example: F of H of 2
- F(H(2)) means evaluate H(2) first, then input that result into F.
- H(2) = 1; then F(1) = (1)² - 1 = 1 - 1 = 0.
- Alternative approach: substitute H(2) value directly into F.
Triple Composition Example
- Compose three functions: H(G(F(2))).
- Step 1: F(2) = 2² - 1 = 4 - 1 = 3.
- Step 2: G(3) = 4.
- Step 3: H(4) = -1.
- Final result: H(G(F(2))) = -1.
Key Terms & Definitions
- Function Composition — The process of applying one function to the result of another function, written as F(G(x)).
- Composite Function — A function created by combining two or more functions, such as F(G(x)).
- Input — The initial value entered into the innermost function.
- Output — The final result after all functions have been applied.
Action Items / Next Steps
- Practice evaluating composite functions with different function definitions.
- Review function notation and mapping for better understanding.