FUNCTIONS & RELATIONS

1 2 3 4 5 6 7 8 9 10–1–2–3–4–5–6–7–8–9–10 x

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–1

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FUNCTIONS & RELATIONS The Coordinate Plane

x axis

QUADRANTII

(– , +)

QUADRANTI

(+ , +)

 

QUADRANTIII

(– , –)  

QUADRANTIV

(+ , –)  

y axis 

The Origin: (0, 0)

(x, y) = ordered pair

Abscissa – The first coordinate of an ordered pair of real numbers that is assigned to a point on the coordinate plane. (x – coordinate)Ordinate – The second coordinate of an ordered pair of real numbers that is assigned to a point on the coordinate plane. (y – coordinate)

Identify which point is…

1 2 3 4 5 6 7 8 9 10–1–2–3–4–5–6–7–8–9–10 x

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• A

• C •

B• D •

E• F

• G

Located in the Fourth Quadrant. Located on the x – axis.

Located on the y – axis.

Located in the Second Quadrant. Located at (6, –6).

Relation:A relation is any set of ordered pairs.

Consider the set {(0, 2), (– 4, 3), (–3, –2), (2, –1)}.

• The set of all first coordinates of the ordered pairs is called the domain of the relation and is often named D.

• The set of all second coordinates of the ordered pairs is called the range of the relation and is often called R.

Domain and Range• Therefore, in the relation {(0, 2), (– 4, 3), (–3, –2), (2, –1)}…

D = {0, –4, –3, 2} R = {2, 3, –2, –1}

The domain is the set of independent variable, x’s, and the range is the set of dependent variables, y’s.

Relation:There are many ways to represent a relation.1) Set Notation using braces: {(0, 2), (– 4, 3), (–3, –2), (2, –1)}.

2) Table:

3) Mapping Diagram:

Functions:If no two ordered pairs in a relation have the same first coordinate, the relation is called a function.

A function is the pairing between two sets of numbers in which every element in the first set is paired with exactly one element of the second set. In other words…

In a function, for every x there is only one y.

Functions:

Real Life Examples:A relation can be a relationship between sets of information. For example, consider the set of all of the people in your Algebra class and the set of their heights is a relation. The pairing of a person and his or her height is a relation.

In relations and functions, the pairs of names and height are ordered, which means one comes first and the other comes second. These two sets could be ordered with the person first {student, height} or the person last {height, student}.

While all functions are relations, since they pair information, not all relations are functions.

Real Life Examples:In the student/height example, the relation {student, height} is a function because every person has only one height at any given point in time. The relation {height, student} is not a function because for every height there might be many students that are that tall.Other examples of relations that are functions: {person, social security number}{dog, tail}{HMS student, homebase}

Other examples of relations that are not functions: {person, telephone number}{HMS student, club or team}{dog, feet}

The Vertical Line TestGiven the graph of a relation, if you can draw a vertical line that crosses the graph at more than one point, then the relation is not a function.

This relation IS as function!

The Vertical Line TestGiven the graph of a relation, if you can draw a vertical line that crosses the graph at more than one point, then the relation is not a function.

This relation IS NOT a function!

Function NotationA rule for a relation is said to define the relation. Given a rule and a domain (the independent variable(s) x) for a relation or function, it is possible to determine all the ordered pairs that form the relation, which will determine the range (the dependent variable(s) y) of the relation.

EXAMPLE: Find the range, R, of the relation defined by the rule 2x + 1, if the domain, D, = {0, 1, 2, – 1, –2}

Function NotationEXAMPLE: Find the range, R, of the relation defined by the rule 2x + 1, if the domain, D, = {0, 1, 2, – 1, –2}

An expanded “T” table can be used to organize the data:

Therefore, R = {1, 3, 5, – 1, –5}

Function NotationA function is named by a single letter, such as f, F, or g.

For example, the function defined by the rule 2x + 1 may be called f and can specified in two ways:

Function Notation: f(x) = 2x + 1 “f of x” is used to denote the specific value of the function that is paired with the number x.

Arrow Notation: f:x 2x + 1

“The function f that pairs a number x with the number 2x + 1”

Function Notation

Example #1: State the domain and range of the function {(0, 2), (2, 4), (4, 6), (6, 8)}.

D = {0, 2, 4, 6 }

{(0, 2), (2, 4), (4, 6), (6, 8)}.

R = { 2, 4, 6, 8 }

Function Notation

Example #2: Given D = {–2, –1, 0, 1, 2 }, determine range of the function f(x) = x – 1.

R = {–3, –2, –1, 0, 1}

 * * Note: the order of the range should match the order of the domain. There is a one-to-one correspondence between each domain value and each range value.

Function NotationExample #3: Find each of the following for the function f(x) = 2x – 4

Function NotationExample #3: Find each of the following for the function f(x) = 2x – 4

Function NotationExample #4:Find each of the following for the function g(x) = 3x2 + 2x + 4

1) g(0) = 4

3) g(–1) = 5

2) g(1) = 9

Function NotationExample #4:Find each of the following for the function g(x) = 3x2 + 2x + 4

4) g = 5

5) g = 3

6) g = 4