Drill #16List the relation (set of ordered pairs) and the

domain and range of the following mapping:

1.

Graph the following relation, state the domain and identify if it a function

2. {(3,-4), (-2, 4), (-3,4)} 3. {(1,-2), (-1, 2), (-1,4)}

0234

x y

-1

2-1 Relations and Functions

Objective: To graph a relation, state its domain and range, and determine if it is a function, and to find values of functions for given elements in a domain.

Cartesian Coordinate PlaneCartesian Coordinate Plane: Composed of an

x-axis (horizontal) and y-axis (vertical) which meet at the origin and divide the plane into four quadrants.

x – axis: The horizontal axis in the coordinate plane.

y – axis: The vertical axis in the coordinate plane.

origin: The point where the x-axis meets the y-axis corresponding the coordinate (0,0)

The Coordinate Plane

Quadrant I

( + , + )

Quadrant II

( - , + )

Quadrant III

( - , - )

Quadrant IV

( + , - )

x

y

(0,0) Origin

Relation, Domain, and Rangerelation: A set of ordered pairs.

domain: The set of all the x – coordinates (the 1st numbers) of a relation. For a function, it’s the set of all possible values of x.

range: The set of all the y – coordinates (the 2nd numbers) of a relation. For a function, it’s the set of all possible values of y.

Example: Name the domain and range of the following relation: { (-1, 2), (-1, 3), (1, 3) }

Mapping

mapping: Shows how each element of the domain is paired with each element of the range.

Example: { (-1, 2), (-1, 3), (-1, 4) }

-1234

D R

Functions

function: A special type of relation in which each element of the domain is paired with exactly one element of the range. (no x- values are repeated)

NOTE : In a function, every x – value (input) has exactly one y – value (output).

discrete function: a function that consists of points that are not connected.

Continuous Functions

continuous functions: A function that can be graphed with a line or a smooth curve and has a domain with an infinite number of elements.

x

y

(0,0) Origin

1

-1

Vertical Line Test

Vertical Line Test: If a vertical line intersects a graph at more than one point then the relation is not a function.

Pass a pencil vertically over a graph. If the pencil ever touches more than one point at a time on the graph, then the graph fails the vertical line test, and is not a function.

Every x – value must have a unique y –value.

Mapping: Classwork5

Identify the domain and range of each mapping. State whether or not each is a function:

A. B.

-1 2

3

D R

01

-3 -1

35

RD4

Classwork

Draw a mapping of the following relations.

State the a) Domain, b) Range of each set.

A) {(1, 2), (1, 3), (1, 4)}

B) {(2, 3), (-1, 3), (1, -3)}

One to One*

One to One Functions: A function such that each element of the domain is paired with exactly one unique element in the range.

One to one One to one Not one to one

D R D R D R

-1 2

3

4

01

-3 -1

3

45

2

3

4

-1

3

Onto*

Onto Functions: A function such that each element of the range is paired with exactly one unique element in the domain.

Onto Onto Not onto

(not a function)

D R D R D R

-1 2

3

4

01

-3 -1

3

45

2

4

-1

3

5

One to One and Onto*

One to one and onto: Each element of the domain is paired with a unique range value, and all range values are paired with a domain value.

One to one One to one Not one to oneand onto not onto Onto

D R D R D R

-1 2

3

4

01

-3

4

-1

3

4

2

4

6

-1

3

Horizontal Line Test: One to One Test

Horizontal Line Test: If a horizontal line intersects a graph at more than one point then the relation is not one to one.

Pass a pencil horizontally over a graph. If the pencil ever touches more than one point at a time on the graph, then the graph fails the vertical line test, and is not a function.

Every y – value must have a unique x –value.

Evaluating Functions

Evaluate the following for the given functions, giving your answer in function notation:

Ex1: f(1) Ex5: g(-1)

Ex2: f(-½) Ex6: g(¼)

Ex3: f(a) Ex7: g(a)

Ex4: f(2x+1) Ex8: g(-t)

52)(,43)( 2 xxgxxf

Evaluating functionsGraph the following function determine the a.)

Domain and Range, b.) whether the equation is a function, whether is one-to-one, onto, both or neither, c.) whether it is discrete or continuous.

x f(x)

-3

-2

-1

0

1

2

3

32)(:1 xxfex

Evaluating functionsGraph the following function determine the a.)

Domain and Range, b.) whether the equation is a function, whether is one-to-one, onto, both or neither, c.) whether it is discrete or continuous.

x f(x)

-3

-2

-1

0

1

2

3

3)(:2

1)(:1 2

xfex

xxfex