Section 2.1 Functions

1. Relations

A relation is any set of ordered pairs

Definition

DOMAINRANGE

independent variable dependent variable

2. Definition of a function

Function: a relation where:• each element in domain corresponds to• EXACTLY one element in the range.

{(2,6),(-3,6),(4,9),(2,10)}

Definitions

Examples of functions.

Domain: set of inputs for a functionRange: set of outputs for a function.

82 2 xxy

3. Functions as Equations

Determine if an equation is a function.

1. Solve for y.

2. If each x is associated with a unique y, then function

Goal:

Method 1: Algebraically

Method 2: Graphically – Vertical Line Test

1. Graph the equation.

2. If no vertical line intersects the graph more than

once, then function

3. Test if a relation is a function:

Test Algebraically or Test with

Vertical Line Test2xy

x y 2

16)4()2( 22 yx

1)

2)

3)

tells us to apply the rule to a “number” xargument (independent variable)

Function Notation Equation Notation

4. Function Notation

)(xf)(xf

f

)3(f

function name

32)( 2 xxxf 322 xxy

Output is in Range

5. Function as a machine

Example

Inputfrom Domain

)(xfx12)( 2 xxf

)0(f

)1( xf

)( xf

)2( xf

Example: 1. Subtract

2. Add

3. Multiply

4. Divide

6. Constructing Functions

)()())(( xfxgxfg Algebraic combinations of functions to form a new function.

)()())(( xgxfxgf

)()())(( xgxfxfg

)(

)()(

xg

xfx

g

f

Study Tip: Cannot split the argumentComposition Functions are not multiplication ))(( xgf

)()()( hfxfhxf

4)(

13)(2

xxf

xxg

7. Domain

Example: State the domain : 1

1)(

x

xf

Definition

Are there any x-values that would make f(x) not real?

Set-builder notation: Domain: 1| xx

Interval notation Domain: ,11,

In words: Domain: 1x

Domain: The largest set of real numbers for which

f(x) is a real number

7. a) Examples: Finding the Domain

f (x)x 2 3x 1

f (x)2x

3x 5x 2

85)( xxf

Polynomial

Rational

Radical (Square Root)

Function Type DomainExample

9. Difference Quotient

The Difference Quotient:

f (x h) f (x)h

Tells us the rate of change of a function.

1)

2) 132)( 2 xxxf

43)( xxf

tells us to apply the rule for to a “number” x is a symbol for the function (could be any letter)

is the argument (independent variable)

The value of is the y coordinate on the graph.

2.2 The Graph of )(xf

)(xff

f

x

)(xf )(,),( xfxyx