Ch 5 Functions Chapter 5: Functions Section 5.1: Examples {1,2,9,12,14} Section 5.4: Examples {2,3,4,5,6,7,9,10}

5.1 Functions

Review A relation between two variables x and y is a set of ordered pairs An ordered pair consist of a x and y-coordinate A relation may be viewed as ordered pairs, mapping design, table, equation, or written in sentences x-values are inputs, domain, independent variable y-values are outputs, range, dependent variable

Example 1 What is the domain? {0, 1, 2, 3, 4, 5} What is the range? {-5, -4, -3, -2, -1, 0}

Example 2 What is the domain? {4, -5, 0, 9, -1} What is the range? –5 9 –1 Input –2 7 Output What is the domain? {4, -5, 0, 9, -1} What is the range? {-2, 7}

Is a relation a function? What is a function? According to the textbook, “a function is…a relation in which every input is paired with exactly one output” 6

Function, a special type of relation

Is a relation a function? Focus on the x-coordinates, when given a relation If the set of ordered pairs have different x-coordinates, it IS A function If the set of ordered pairs have same x-coordinates, it is NOT a function Y-coordinates have no bearing in determining functions

YES Example 3 Is this a function? Hint: Look only at the x-coordinates :00 Is this a function? Hint: Look only at the x-coordinates YES

NO Example 4 Is this a function? Hint: Look only at the x-coordinates :40 Example 4 Is this a function? Hint: Look only at the x-coordinates NO

Choice 1 Example 5 Choice One Choice Two :40 Example 5 Which mapping represents a function? Choice One Choice Two 3 1 –1 2 2 –1 3 –2 Choice 1

Example 6 Which mapping represents a function? A. B. B

Function Notation f(x) means function of x and is read “f of x.” f(x) = 2x + 1 is written in function notation. The notation f(1) means to replace x with 1 resulting in the function value. f(1) = 2x + 1 f(1) = 2(1) + 1 f(1) = 3

Functions Injections, Surjections and Bijections Let f be a function from A to B. Definition: f is one-to-one (denoted 1-1) or injective if preimages are unique. Note: this means that if a  b then f(a)  f(b). Definition: f is onto or surjective if every y in B has a preimage. Note: this means that for every y in B there must be an x in A such that f(x) = y. Definition: f is bijective if it is surjective and injective (one-to-one and onto).

Functions A B a X b Y c Z d Surjection but not an injection

Functions A B A B a a V V b b W W c c X X d d Y Y Injection & a surjection, hence a bijection Z Injection but not a surjection

Functions Inverse Functions Definition: Let f be a bijection from A to B. Then the inverse of f, denoted f-1, is the function from B to A defined as f-1(y) = x iff f(x) = y

Functions Definition: Let S be a subset of B. Then f-1(S) = {x | f(x)  S} Note: f need not be a bijection for this definition to hold. Example: Let f be the following function: A B a X f-1({Z}) = {c, d} f-1({X, Y}) = {a, b} b Y c Z d

Functions Composition Definition: Let f: B C, g: A B. The composition of f with g, denoted fg, is the function from A to C defined by f  g(x) = f(g(x))

A g B f C a V h Examples: b W i c X j d Y A fg C a h b i c j d

Examples

Examples

Examples

Examples

Examples