Overview
This lecture explains the concept of function composition in mathematics, covers the notation, properties, and walks through multiple worked examples including step-by-step solutions.
Introduction to Function Composition
- Function composition combines two functions, f(x) and g(x), to form a new function.
- The result of composing f and g depends on the order of operations.
- It is denoted as (f ∘ g)(x) or "f composition g of x," meaning f(g(x)).
- Composition can involve more than two functions, resulting in nested or layered operations.
Notation and Order
- (f ∘ g)(x) = f(g(x)): g(x) is calculated first, then input into f.
- (g ∘ f)(x) = g(f(x)): f(x) is calculated first, then input into g.
- The order of composition affects the final result; (f ∘ g)(x) ≠(g ∘ f)(x) in general.
Key Properties of Function Composition
- Non-Commutative: Composition order matters; (f ∘ g)(x) is not the same as (g ∘ f)(x).
- Associative: (f ∘ (g ∘ h))(x) = ((f ∘ g) ∘ h)(x), grouping does not change the result.
- Identity Function: Composing any function with the identity function i(x) = x leaves the original function unchanged.
Worked Examples
Given:
f(x) = 2x + 3, g(x) = x² – 4, h(x) = x – 2, i(x) = x (identity)
- (f ∘ g)(x) = f(g(x)) = 2(x² – 4) + 3 = 2x² – 8 + 3 = 2x² – 5
- (g ∘ f)(x) = g(f(x)) = (2x + 3)² – 4 = 4x² + 12x + 9 – 4 = 4x² + 12x + 5
- (f ∘ g ∘ h)(x) = f(g(h(x))) = f((x–2)² – 4) = 2(x² – 4x + 4 – 4) + 3 = 2x² – 8x + 3
- (h ∘ g ∘ f)(x) = h(g(f(x))) = h((2x + 3)² – 4) = (4x² + 12x + 5) – 2 = 4x² + 12x + 3
- (f ∘ i)(x) = f(x) = 2x + 3; (i ∘ f)(x) = f(x)
- Calculating the value at a point: (f ∘ g)(2) = f(g(2)) = f(0) = 3
Key Terms & Definitions
- Function Composition — Combining two functions so one’s output becomes the other’s input, written as (f ∘ g)(x).
- Identity Function — Function i(x) = x, which leaves other functions unchanged when used in composition.
- Non-Commutative — Property where changing the order changes the result.
- Associative — Grouping does not affect the outcome; (f ∘ (g ∘ h)) = ((f ∘ g) ∘ h).
Action Items / Next Steps
- Practice problems involving function composition with different function formulas.
- Review associativity and non-commutativity in composition.
- Prepare for exercises on computing values of composed functions at given x-values.