(2-1) Relations and Functions

Cartesian Coordinate Plane

Def:

Composed of the

x-axis (horizontal) and the y-axis (vertical) which meet at the origin (0,0).

Relation

Def: Set of ordered pairs

Example: {(3,2),(4,1),(7,3),

(1,2),(6,3),(5,6)}

Quadrants

Def:

The Cartesian

coordinate plane is divided into four quadrants (parts), which can be written in the form (x,y).

Domain

Def:

Domain of a relation is the set of all x-values in a set of ordered pairs.

Domain

Def:

Domain of a relation is the set of all x-values in a set of ordered pairs.

x,y x,y x,y x,y x,y x,y

Example: {(3,2),(4,1),(7,3),(1,2),(6,3),(5,6)}

Domain

Def:

Domain of a relation is the set of all x-values in a set of ordered pairs.

x,y x,y x,y x,y x,y x,y

Example: {(3,2),(4,1),(7,3),(1,2),(6,3),(5,6)}

Domain: { 3, 4, 7, 1, 6, 5 }

Range

Def:

The range of a relations is the set of all

y-values in a set of ordered pairs.

x,y x,y x,y x,y x,y x,y

Example: {(3,2),(4,1),(7,3),(1,2),(6,3),(5,6)}

Range: { 2, 1, 3, 2, 3, 6 }

Range: { 2, 1, 3, 6} , you don’t repeat

Mapping

Def:

Shows how the member of the domain is paired with each member of the range.

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X Y

f(x)

Example: {(3,2),(4,1),(7,3),(1,2),(6,3),(5,6)}

x,y x,y x,y x,y x,y x,y

Function

Def:

A function is a special type of relation in which each element of the domain is paired with exactly one element of the range.

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42

X Y

f(x)

Example: {(2,4),(1,2),(5,4),(4,2)}

x,y x,y x,y x,y

Not a Function

241

134

X Yf(x)

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

x,y x,y x,y x,y

This is not a functionbecause 2 goes to 1 and 3, which contradicts the definitionof a function.

Discrete Vs. Continuous

Discrete is a relation in which the Domain is a set of individual points.

Continuous is when the domain of a relation has an infinite number of elements and the relation can be graphed with a line or smooth curve.

Independent Variable

When an equation represents a function, the variable, usuallyx, whose values make up the domain is called the Independent Variable.

y = 3x+4

Independent Variable

Dependent Variable

The other variable, usually y, is called the Dependent Variablebecause its values depend on x.

y = 3x+4

Independent Variable

Dependent Variable

Vertical Line Test

If no vertical line intersects a graph in more than one point, the graph represents a function.

Vertical Line Test

If no vertical line intersects a graph in more than one point, the graph represents a function.

Thus this is not a function becausethe vertical line crosses the graph through more than one point.

Evaluating a Function

Given ,find each value.

(a) f (-3)

Take original function

Substitute

Simplify= 9+2=11

Evaluating a Function

Given ,find each value.

(b) f (3z)

Take original function

Substitute

Simplify

Class WorkDetermine whether each relation is a function. Write yes or no.

Discrete or ContinuousGraph each relation or equation and find the domain and rang. Next determineif the relation is discrete or continuous. Then determine whether the relation or equation is a function.

Evaluate