Foundations Functions Domain and Range Relations Functions Relation

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Foundations Functions, Domain, and Range

Foundations Functions, Domain, and Range

Relations & Functions Relation: a set of ordered pairs Domain: the set of x-coordinates

Relations & Functions Relation: a set of ordered pairs Domain: the set of x-coordinates Range: the set of y-coordinates When writing the domain and range, do not repeat values.

Relations and Functions Given the relation: {(2, -6), (1, 4), (2, 4), (0, 0),

Relations and Functions Given the relation: {(2, -6), (1, 4), (2, 4), (0, 0), (1, -6), (3, 0)} State the domain: D: {0, 1, 2, 3} State the range: R: {-6, 0, 4}

Relations and Functions • Relations can be written in several ways: ordered pairs, table,

Relations and Functions • Relations can be written in several ways: ordered pairs, table, graph, or mapping. • We have already seen relations represented as ordered pairs.

Ordered pair Table {(3, 4), (7, 2), (0, -1), (-2, 2), (-5, 0), (3,

Ordered pair Table {(3, 4), (7, 2), (0, -1), (-2, 2), (-5, 0), (3, 3)} Mapping 3 4 7 2 0 -1 -2 0 -5 3 Graph

Functions • A function is a relation in which the members of the domain

Functions • A function is a relation in which the members of the domain (x-values) DO NOT repeat. • So, for every x-value there is only one y-value that corresponds to it. • y-values can be repeated.

Functions • Discrete functions consist of points that are not connected. • Continuous functions

Functions • Discrete functions consist of points that are not connected. • Continuous functions can be graphed with a line or smooth curve and contain an infinite number of points.

Do the ordered pairs represent a function? {(3, 4), (7, 2), (0, -1), (-2,

Do the ordered pairs represent a function? {(3, 4), (7, 2), (0, -1), (-2, 2), (-5, 0), (3, 3)} No, 3 is repeated in the domain. {(4, 1), (5, 2), (8, 2), (9, 8)} Yes, no x-coordinate is repeated.

Write the domain and range in {set notation}

Write the domain and range in {set notation}

Graphs of a Function Vertical Line Test: If a vertical line is passed over

Graphs of a Function Vertical Line Test: If a vertical line is passed over the graph and it intersects the graph in exactly one point, the graph represents a function.

Does the graph represent a function? Name the domain and range. x Yes D:

Does the graph represent a function? Name the domain and range. x Yes D: all real numbers R: all real numbers y x y Yes D: all real numbers R: y ≥ -6

Does the graph represent a function? Name the domain and range. x No D:

Does the graph represent a function? Name the domain and range. x No D: x ≥ 1/2 R: all reals y x y No D: all reals R: all reals

Does the graph represent a function? Name the domain and range. x Yes D:

Does the graph represent a function? Name the domain and range. x Yes D: all reals R: y ≥ -6 y x y No D: x = 2 R: all reals

Function Notation • When we know that a relation is a function, the “y”

Function Notation • When we know that a relation is a function, the “y” in the equation can be replaced with f(x). • f(x) is simply a notation to designate a function. It is pronounced ‘f’ of ‘x’. • The ‘f’ names the function, the ‘x’ tells the variable that is being used.

Introducing…… Interval Notation!

Introducing…… Interval Notation!

List the domain and range in interval notation x Yes D: R: y x

List the domain and range in interval notation x Yes D: R: y x y Yes D: R:

List the domain and range in interval notation x No D: R: y x

List the domain and range in interval notation x No D: R: y x y No D: R:

List the domain and range in interval notation x Yes D: R: y x

List the domain and range in interval notation x Yes D: R: y x y No D: R:

List the domain and range in interval notation

List the domain and range in interval notation

Order of Operations (What we use when evaluating the expressions) • • • P

Order of Operations (What we use when evaluating the expressions) • • • P E M D A S *Multiply/Divide and Add/Subtract from left to right! Please excuse my dear Aunt Sally?

Examples

Examples

Examples • Evaluate the expression.

Examples • Evaluate the expression.