Overview
This lecture covers algebraic operations with functions—including addition, subtraction, multiplication, division, and composition—and demonstrates how to perform and evaluate these operations with examples.
Basic Function Operations
- Functions provided: ( f(x) = 3x - 5 ), ( g(x) = x - 2 ), ( h(x) = 3x^2 - 11x + 10 ).
- The sum of two functions: ( (f+g)(x) = f(x) + g(x) = (3x - 5) + (x - 2) = 4x - 7 ).
- The quotient of two functions: ( (h/g)(x) = h(x) / g(x) = (3x^2 - 11x + 10)/(x - 2) ).
- Factoring numerator ( 3x^2 - 11x + 10 = (x - 2)(3x - 5) ), so ( (h/g)(x) = 3x - 5 ) after cancelling ( x-2 ).
- Product evaluated at a value: ( (f \cdot g)(3) = f(3) \times g(3) = 4 \times 1 = 4 ).
Function Composition
- Function composition: ( f \circ g )(x) means ( f(g(x)) ).
- New example functions: ( f(x) = x^2 + 3x ), ( g(x) = 4x - 1 ).
- ( (f \circ g)(0) = f(g(0)) = f(-1) = (-1)^2 + 3(-1) = 1 - 3 = -2 ).
- ( (g \circ f)(0) = g(f(0)) = g(0) = 4 \times 0 - 1 = -1 ).
- Composition is not commutative: ( f \circ g (0) \ne g \circ f (0) ).
- General form: ( (f \circ g)(x) = f(4x - 1) = (4x-1)^2 + 3(4x-1) ) (expansion optional).
- ( (g \circ f)(x) = g(x^2 + 3x) = 4(x^2 + 3x) - 1 = 4x^2 + 12x - 1 ).
Key Terms & Definitions
- Function — A rule that assigns each input exactly one output.
- Sum of functions — ( (f+g)(x) = f(x) + g(x) ).
- Product of functions — ( (fg)(x) = f(x) \times g(x) ).
- Quotient of functions — ( (f/g)(x) = f(x)/g(x) ), ( g(x) \ne 0 ).
- Composition of functions — ( (f \circ g)(x) = f(g(x)) ), order matters (not commutative).
Action Items / Next Steps
- Expand and simplify ( (f \circ g)(x) = (4x-1)^2 + 3(4x-1) ) for homework.
- Practice function operations using other function pairs and values.