- Section 6.4
- Suppose f(x) = x2 - 3, g(x) = sinx + cosx
- What is g ○ f?
- “plug f in for x in g”
- dom(f ○ g) = dom(f)
- codom(f ○ g) = codom(g)
- So g ○ f = sin(x2 - 3) + cos(x2 - 3)
- What is f ○ g?
- (sinx + cosx)2 - 3
- What is g ○ f?
- Find functions f and g such that h(x) = sin( (x+1)x+1 ) = f ○ g
- f(x) = sin( xx ), g(x) = x+1
- Or f(x) = sin x, g(x) = (x+1)(x+1)
- (Several different functions can compose to the same result)
- Is composition a function?
- Call it C
- Think of A → B in “f : A → B” as the set of functions from A
to B
- You can form sets out of very complicated things, e.g., functions
- dom(C) = (A→B) × (B→D)
- codom(C) = A→D
- A concrete example:
- f : R → R, f(x) = x2
- g : R → Z, g(x) = ⌊x⌋
- g ○ f = ⌊x2⌋
- Inverses of functions
- Read section 6.5