University:
Amherst College
Course:
MATH 220 | Mathematical Reasoning and Proof
Academic year:

2020
Views:

301

Pages:

2
Author:

catfever701r

Some Basic Concepts about Functions Please read this handout carefully and ask questions in office hours about any aspect you don’t fully understand. §1: Basic Definitions Definition. Let A and B be sets. • A function f : A → B means that for every a ∈ A, there is a unique f (a) ∈ B. We call A the domain of f . • Two functions f, g : A → B are equal, written f = g, if f (a) = g(a) for all a ∈ A. Definition. Given functions f : A → B and g : B → C, the composition g ◦ f : A → C is the function defined by (g ◦ f )(a) = g(f (a)) for all a ∈ A. Example. Let f : R → R be f (x) = x2 and g : R → R be g(x) = sin(x). Then (g ◦ f )(x) = sin(x2 ) and (f ◦ g)(x) = sin(x)2 = sin2 (x). §2: One-to-One and Onto Definition. A function f : A → B is: • one-to-one or 1-1 or injective if for all elements a, b ∈ A, a 6= b =⇒ f (a) 6= f (b). • onto or surjective if for every b ∈ B there is at least one a ∈ A such that f (a) = b. Facts about One-to-One. • One-to-one is equivalent to saying that f : A → B is one-to-one if for all elements a, b ∈ A, f (a) = f (b) =⇒ a = b. Example. sin : R → R is not one-to-one but sin : [−π/2, π/2] → R is. Facts about Onto. • The range of a function f : A → B is the set R(f ) = {f (a) | a ∈ A} ⊆ B. • A function f : A → B is onto if and only if R(f ) = B, i.e., its range is as big as possible. Be sure you understand this. Example. sin : R → R is not onto but sin : R → [−1, 1] is onto since [−1, 1] is the range of sin. Be sure you understand this. 2 §3: Inverse Functions and Bijections Definition. • Given any set A, the identity function IA : A → A is the function defined by IA (a) = a for every a ∈ A. • Functions f : A → B and g : B → A are called inverse functions if g ◦ f = IA and f ◦ g = IB . • A function is bijective if it is one-to-one and onto. Facts about Inverse Functions. • Functions f : A → B and g : B → A are inverse functions if and only if g(f (a)) = a f (g(b)) = b for all a ∈ A for all b ∈ B. • If f : A → B has an inverse function, then the inverse is unique and is denoted f −1 : B → A. • If f : A → B has an inverse function f −1 , then f is the inverse of f −1 , i.e., (f −1 )−1 = f . Examples of Inverse Functions. • Let R>0 = (0, ∞) be the set of positive real numbers. Then the functions ex = exp(x) and ln(x) from calculus give inverse functions since exp : R → R>0 , ln : R>0 → R exp(ln(x)) = eln(x) = x for all x ∈ R>0 ln(exp(x)) = ln(ex ) = x for all x ∈ R. • sin : [−π/2, π/2] → [−1, 1] and sin−1 = arcsin : [−1, 1] → [−π/2, π/2] are inverse functions. You studied these functions in calculus. Facts about Bijections and Inverse Functions • f : A → B has an inverse function if and only f is bijective. • If f : A → B and h : B → C are bijections, then the composition h ◦ f : A → C is a bijection. Furthermore, its inverse function is given by (h ◦ f )−1 = f −1 ◦ h−1 . Example of a Bijection. The squaring function f : [0, ∞) → [0, ∞) defined by f (x) = x2 is bijective (do you see why?). Hence it has an inverse function f −1 :√[0, ∞) → [0, ∞), which is clearly the square root function, i.e., f −1 (x) = x. The “Facts” stated above are theorems whose proofs we will skip. Math 220 covers this material in detail. See also page 156 of the linear algebra text.

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