7.3 Composition of Functions

Definition: Composition of f and g

Let f: X → Y and g: Y^' → Z be functions with the property that the range of f is a subset of the domain of g. Define a new function g ∘ f: X → Z as follows:

• (g ∘ f)(x) = g(f(x)) for each x ∈ X

where g ∘ f is read "g circle f" and g(f(x)) is read "g of f of x".

Theorem 7.3.1 Composition with an Identity Function

If f is a function from a set X to a set Y, and I_X is the identity function on X, and I_Y is the identity function on Y, then

• f ∘ I_X = f
• I_Y ∘ f = f

Theorem 7.3.2 Composition of a Function with its Inverse

If f: X → Y is a one-to-one and onto function with inverse function f^-1: Y → X, then

• f^-1 ∘ f = I_X
• f ∘ f^-1 = I_Y

Theorem 7.3.3 Composition of One-to-One Functions

If f: X → Y and g: Y → Z are both one-to-one functions, then g ∘ f is one-to-one.

Theorem 7.3.4 Composition of Onto Functions

If f: X → Y and g: Y → Z are both onto functions, then g ∘ f is onto.

In Class Exercises

• 3. F(x) = x³ and G(x) = x - 1, for each real number x. What is F ∘ G? What is G ∘ F? Are these equal?
• 10. Define F: Z → Z and G: Z → Z by the rules F(n) = 2n and G(n) = floor(n/2) for every x ∈ Z.
  • Find (G ∘ F)(3)
  • Find (F ∘ G)(3)
  • Is G ∘ F = F ∘ G? Explain.
• 12. F: R → R and F^-1: R → R are defined by F(x) = 3x + 2 and F^-1(y) = (y-2)/3 for every x, y ∈ R. Check that both compositions give the identity function.
• 16. Prove Theorem 7.3.1(b): If f is a function from a set X to a set Y, then I_Y ∘ f = f, where I_Y is the identity function on Y.