ICS 6B Chapter 4

4.1: Definition of functions

Function: Maps elements from one set to another using some sort of operation
- Can be represented as a set and as a subset of X × Y such that for every x ∈ X, there is exactly one y ∈ Y for which (x, y) ∈ f
- f: X -> Y is used to show that f is the function that maps set X to Y
- Set X is the domain of the function while set Y is the target or co-domain of the function
- “f maps x to y” can be written as (x, y) ∈ f and can also be denoted as f(x) = y
- If a function maps an element of the domain to either 0 elements or >1 element, then it is not well-defined
- An element is in the range of f if there is an x ∈ X such that (x, y) ∈ f
  - Alternate definition: Range of f = { y: (x, y) ∈ f, for some x ∈ X }
A function that has a finite domain can be shown graphically as an arrow diagram
- Example:
- Non well-defined function example:
A function definition is incomplete without specifying the domain and target
- Correct: g: R → R, where g(x) = |x|
- Incorrect: g(x) = |x|

Floor function: Function that maps a real number to its closest integer, rounded down
- Definition: floor: R → Z, where floor(x) = the largest integer y such that y ≤ x.
- Notated as such:
Ceiling function: Function that maps a real number to its closest integer, rounded down
- Definition: ceiling: R → Z, where ceiling(x) = the smallest integer y such that x ≤ y.
- Notated as such:

A function is one-to-one or injective if each value of x maps to a unique value of y
- x₁ ≠ x₂ implies that f(x₁) ≠ f(x₂)
A function is onto or surjective if the range of f is equal to its target
- For every y ∈ Y, there is an x ∈ X such that f(x) = y
A function is bijective if it is both onto and one-to-one
- Such functions are called bijections or one-to-one correspondences
Can infer the sizes, or cardinality, of the domain and target sets depending on the function’s properties
- If f: D → T is onto, then for every element in the target, there is at least one element in the domain: |D| ≥ |T|.
- If f: D → T is one-to-one, then every element in the domain maps to a unique element in the target: |D| ≤ |T|.
- If f: D → T is a bijection, then f is one-to-one and onto: |D| ≤ |T| and |D| ≥ |T|, which implies that |D| = |T|.
- Visual:

Challenge Activity is good to practice onto vs. one-to-one

The inverse of a function is found by reversing the order of its tuples; that is, f-1 = { (y, x) : (x, y) ∈ f }.
- Inverses may not be well-defined, so some functions don’t have inverses
- Example:
- A function has an inverse if and only if it is a bijection
- Analytically solving for the inverse of a function with an infinite domain:

A composition of functions is when a function is applied to another function
- Example: f(g(x)), or (f ο g)(x)
- Visual example:
The identity function (denoted by I) takes a set and maps each of its elements onto itself
- If a function f has an inverse, then f composed with its inverse is an identity function
- Mathematical definitions:
  - (f o f^-1)(x) = I_f
  - Let f: A → B be a bijection. Then f^-1 ο f = I_A and f ο f^-1 = I_B.

Exponential function and logarithmic function are featured
- Properties of exponents:
- Properties of logarithms:
A function is strictly increasing if, for each increasing value of x, f(x) is bigger than the last
- if whenever x₁ < x₂, then f(x₁) < f(x₂)
A function is strictly decreasing if, for each increasing value of x, f(x) is smaller than the last
- if whenever x₁ < x₂, then f(x₁) > f(x₂)
Logarithmic function is strictly increasing