1.3 Graphs of Functions Pre-Calculus. Home on the Range What kind of "range" are we talking about?...
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Transcript of 1.3 Graphs of Functions Pre-Calculus. Home on the Range What kind of "range" are we talking about?...
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