Overview
This lesson covers composite functions, how to form and evaluate them, and the difference between composite and product functions.
Composite Functions: Definition and Notation
- Composite functions combine two functions so that the output of one becomes the input of another.
- Notation: ( f(g(x)) ) means the function ( g(x) ) is inside function ( f ).
- An open circle notation (∘) is used for composite functions, while a closed circle indicates multiplication.
Evaluating Composite Functions: Examples
- For ( f(x) = 3x - 4 ) and ( g(x) = x^2 - 3 ):
- ( f(g(x)) = f(x^2 - 3) = 3(x^2 - 3) - 4 = 3x^2 - 13 )
- ( g(f(x)) = g(3x - 4) = (3x - 4)^2 - 3 )
- ( (3x - 4)^2 = 9x^2 - 24x + 16 ), so ( 9x^2 - 24x + 16 - 3 = 9x^2 - 24x + 13 )
Evaluating Composite Functions at Specific Values
- For ( f(x) = 5x + 2 ) and ( g(x) = x^3 - 4 ):
- To find ( f(g(2)) ):
- ( g(2) = 2^3 - 4 = 8 - 4 = 4 )
- ( f(4) = 5 \times 4 + 2 = 22 )
- To find ( g(f(-1)) ):
- ( f(-1) = 5 \times (-1) + 2 = -3 )
- ( g(-3) = (-3)^3 - 4 = -27 - 4 = -31 )
Key Terms & Definitions
- Composite Function — a function where the output of one function becomes the input of another, written as ( f(g(x)) ).
- Product Function — the result of multiplying two functions, written as ( (f \cdot g)(x) ).
Action Items / Next Steps
- Practice evaluating composite functions for different function pairs and specific values.
- Review the distinction between composite and product functions.