1.3 Graphs of Functions1.3 Graphs of Functions

Pre-CalculusPre-Calculus

Home on the RangeHome on the Range

What kind of "range" What kind of "range" are we talking about?are we talking about?

What does it have toWhat does it have todo with "domain?"do with "domain?"

Are domain and rangeAre domain and rangereally "good fun for the really "good fun for the whole family?"whole family?"

DefinitionsDefinitions

Domain:Domain: Is the set of all first Is the set of all first coordinates (coordinates (xx-coordinates) from -coordinates) from the ordered pairs.the ordered pairs.

Range:Range: Is the set of all second Is the set of all second coordinates (coordinates (yy-coordinates) from -coordinates) from the ordered pairs.the ordered pairs.

DomainDomain

The domain is the set of all possible The domain is the set of all possible inputs into the function { 1, 2, 3, … }inputs into the function { 1, 2, 3, … }

The nature of some functions may The nature of some functions may mean restricting certain values as mean restricting certain values as inputsinputs

RangeRange

{ 9, 14, -4, 6, … }{ 9, 14, -4, 6, … }

The range would be all the possible The range would be all the possible resulting outputsresulting outputs

The nature of a function may restrict The nature of a function may restrict the possible output valuesthe possible output values

Find the Domain and RangeFind the Domain and Range

Given the set of ordered pairs,Given the set of ordered pairs,

{(2,3),(-1,0),(2,-5),(0,-3)}{(2,3),(-1,0),(2,-5),(0,-3)}

DomainDomain

Range Range

Choosing Realistic Domains and Choosing Realistic Domains and RangesRanges

Consider a function used to model a Consider a function used to model a real life situationreal life situation

Let h(t) model the height of a ball as Let h(t) model the height of a ball as a function of timea function of time

What are realistic values for t and for What are realistic values for t and for height?height?

2( ) 16 64h t t t

Choosing Realistic Domains and Choosing Realistic Domains and RangesRanges

By itself, out of context, it is just a By itself, out of context, it is just a parabola that has the real numbers parabola that has the real numbers as domain andas domain and

a limited rangea limited range

2( ) 16 64h t t t

Find the Domain and Range of a Find the Domain and Range of a FunctionFunction

a)a) Find the domain of f(x)Find the domain of f(x)

b)b) Find f(-1)Find f(-1)

c)c) f(2)f(2)

d)d) Find the range of f(x)Find the range of f(x)

*When viewing a graph of a function, realize that solid or open dots *When viewing a graph of a function, realize that solid or open dots on the end of a graph mean that the graph doesn’t extend on the end of a graph mean that the graph doesn’t extend beyond those points. However, if the circles aren’t shown on beyond those points. However, if the circles aren’t shown on the graph it may be assumed to extend to infinity.the graph it may be assumed to extend to infinity.

Domain and RangeDomain and Range

Find the domain Find the domain and range ofand range of

( ) 4f x x

Vertical Line TestVertical Line Test

A set of points in a coordinate plane A set of points in a coordinate plane is the graph of y as a function of x if is the graph of y as a function of x if and only if no vertical line intersects and only if no vertical line intersects the graph at more than one point.the graph at more than one point.

If a vertical line passes through a graph more than once, the graph is not the graph of a function.

Hint:

Pass a pencil across the graph held

vertically to represent a vertical line.

The pencil crosses the graph more than once. This is not a function because there are two y-values for the same

x-value.

The Ups and DownsThe Ups and Downs Think of a function as a roller coaster Think of a function as a roller coaster

going from left to rightgoing from left to right

UphillUphill Slope > 0Slope > 0 IncreasingIncreasing

functionfunction Downhill Downhill

Slope < 0Slope < 0 DecreasingDecreasing function function

19

A function f is increasing on (a, b) if f (x1) < f (x2) whenever x1 < x2.

A function f is decreasing on (a, b) if f (x1) > f (x2) whenever x1 < x2.

Increasing IncreasingDecreasing

Increasing/DecreasingIncreasing/DecreasingFunctionsFunctions

ExampleExample

In the given graph of the function f(x), determine the interval(s) where the function is increasing, decreasing, or constant.

Maximum and Minimum ValuesMaximum and Minimum Values

Local Maximum( f (c2) f (x) for all x in

I •

Absolute Maximum( f (c1) f (x) for all

x)

•

|c2

|c1

I

Maximum and Minimum ValuesMaximum and Minimum Values

Local Minimum( f (c2) f(x) for all x in

I )

•

Absolute Minimum( f (c1) f(x) for all

x)

•

I

c2

||c1 I

Collectively, maximums and minimums are called extreme values.

Approximating a Relative MinimumApproximating a Relative Minimum

Use a calculator to Use a calculator to approximate the approximate the relative minimum relative minimum of the function of the function given by given by

2( ) 3 4 2f x x x

Approximating Relative Minima and Approximating Relative Minima and MaximaMaxima

Use a calculator to Use a calculator to approximate the approximate the relative minimum relative minimum and relative and relative maximum of the maximum of the function given by function given by

3( )f x x x

TemperatureTemperature

During a 24-hour period, During a 24-hour period, the temperature y (in the temperature y (in degrees Fahrenheit) of a degrees Fahrenheit) of a certain city can be certain city can be approximated by the approximated by the model where x model where x represents the time of represents the time of day, with x=0 day, with x=0 corresponding to 6 am. corresponding to 6 am. Approximate the max Approximate the max and min temperatures and min temperatures during this 24-hour during this 24-hour period.period. 3 20.026 1.03 10.2 34,0 24y x x x x

Piecewise Defined FunctionsPiecewise Defined Functions

Sketch the graph of Sketch the graph of by hand.by hand.

2 3, 1( )

4, 1

x xf x

x x

Even functionsEven functions

A function A function ff is an is an even functioneven function if if

for all values of for all values of xx in the domain of in the domain of f.f.

Example: Example: is is even even because because

)()( xfxf

13)( 2 xxf

Odd functionsOdd functions

A function A function ff is an is an odd odd function if function if

for all values of for all values of xx in the domain of in the domain of f.f.

Example: Example: is is odd odd because because

)()( xfxf

xxxf 35)(

Determine if the given functions are even or Determine if the given functions are even or oddodd

23

3

24

)()4

1||)()3

)()2

1)()1

xxxk

xxh

xxg

xxxf

Graphs of Even and Odd functionsGraphs of Even and Odd functions

The graph of an even function is The graph of an even function is symmetric with respect to the symmetric with respect to the x-axis.x-axis.

The graph of an odd function is The graph of an odd function is symmetric with respect to the symmetric with respect to the originorigin..

52.50-2.5-5

5

2.5

0

-2.5

-5

x

y

x

y

Determine if the function is even or odd?

Determine if the function is even or odd?

52.50-2.5-5

5

3.75

2.5

1.25

0

x

y

x

y

52.50-2.5-5

100

50

0

-50

-100

x

y

x

y

Determine if the function is even or odd?

HomeworkHomework

Page 38-41Page 38-41

2-8 even (graphical), 15-18 all, 19-29 2-8 even (graphical), 15-18 all, 19-29 odd, 32-36 even, 44, 48, 59-65 odd, odd, 32-36 even, 44, 48, 59-65 odd, 73-81 odd, 9173-81 odd, 91