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FUNCTIONS
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Learning Outcome
Understand the conceptsof function
Learning outcome
Know the properties offunctions, operation on
functions, and graphs of
functions
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Introduction
are the major tools for describing the real world in mathematical terms
Functions
Eg . The amount of
interest, Iwill depends on
the interest rate, rofferedby the bank
The values of one variable often depend on the values for another
)(:
"ofequals""offunctionais"
)(
rfIEg
xfyxy
xfy
Dependent variable
Independent variable
Notation
Functions is used to describe the dependence of one quantity on another
http://images.google.com.my/imgres?imgurl=http://www.analyzemath.com/calculus/continuity/continuous_1.gif&imgrefurl=http://www.analyzemath.com/calculus/continuity/continuous_functions.html&usg=__hdeWlUUoIXXI71svrUD3lXtBVEs=&h=325&w=423&sz=4&hl=ms&start=4&tbnid=nbH7GGesDcYNwM:&tbnh=97&tbnw=126&prev=/images%3Fq%3Dfunctions%26gbv%3D2%26hl%3Dmshttp://images.google.com.my/imgres?imgurl=http://thumb10.shutterstock.com.edgesuite.net/display_pic_with_logo/1934/1934,1155720118,3/stock-photo-arithmetic-operations-1691873.jpg&imgrefurl=http://www.shutterstock.com/pic-1691873/stock-photo-arithmetic-operations.html&usg=__5862epj1JBXL9eYPrDzh7087zDY=&h=358&w=450&sz=20&hl=ms&start=69&tbnid=GHwV0CJdEnE8kM:&tbnh=101&tbnw=127&prev=/images%3Fq%3Darithmetic%2Boperation%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D54http://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D720http://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D720http://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D720http://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D7207/30/2019 Week1 Functions
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4
Introduction
The area A of a circle depends on its radius r by the equation :
2
)( rrfA
Independent
variable
Dependent
variableSo we say that,
A is a function
of r
Examples
http://images.google.com.my/imgres?imgurl=http://www.analyzemath.com/calculus/continuity/continuous_1.gif&imgrefurl=http://www.analyzemath.com/calculus/continuity/continuous_functions.html&usg=__hdeWlUUoIXXI71svrUD3lXtBVEs=&h=325&w=423&sz=4&hl=ms&start=4&tbnid=nbH7GGesDcYNwM:&tbnh=97&tbnw=126&prev=/images%3Fq%3Dfunctions%26gbv%3D2%26hl%3Dmshttp://images.google.com.my/imgres?imgurl=http://thumb10.shutterstock.com.edgesuite.net/display_pic_with_logo/1934/1934,1155720118,3/stock-photo-arithmetic-operations-1691873.jpg&imgrefurl=http://www.shutterstock.com/pic-1691873/stock-photo-arithmetic-operations.html&usg=__5862epj1JBXL9eYPrDzh7087zDY=&h=358&w=450&sz=20&hl=ms&start=69&tbnid=GHwV0CJdEnE8kM:&tbnh=101&tbnw=127&prev=/images%3Fq%3Darithmetic%2Boperation%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D54http://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D720http://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D720http://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D720http://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D7207/30/2019 Week1 Functions
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Introduction
The velocity v of a ball falling freely in the
Earths gravitational field increases with time,until it hits the ground
So we say that, v
is a function of t
In a bacteria culture , the number n of bacteria
present after 1 hour of growth depends on thenumber no of bacteria present Initially
So we say that,
n is a function no
Examples
http://images.google.com.my/imgres?imgurl=http://www.analyzemath.com/calculus/continuity/continuous_1.gif&imgrefurl=http://www.analyzemath.com/calculus/continuity/continuous_functions.html&usg=__hdeWlUUoIXXI71svrUD3lXtBVEs=&h=325&w=423&sz=4&hl=ms&start=4&tbnid=nbH7GGesDcYNwM:&tbnh=97&tbnw=126&prev=/images%3Fq%3Dfunctions%26gbv%3D2%26hl%3Dmshttp://images.google.com.my/imgres?imgurl=http://thumb10.shutterstock.com.edgesuite.net/display_pic_with_logo/1934/1934,1155720118,3/stock-photo-arithmetic-operations-1691873.jpg&imgrefurl=http://www.shutterstock.com/pic-1691873/stock-photo-arithmetic-operations.html&usg=__5862epj1JBXL9eYPrDzh7087zDY=&h=358&w=450&sz=20&hl=ms&start=69&tbnid=GHwV0CJdEnE8kM:&tbnh=101&tbnw=127&prev=/images%3Fq%3Darithmetic%2Boperation%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D54http://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D720http://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D720http://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D720http://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D7207/30/2019 Week1 Functions
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Introduction
DefinitionDef 1.1.1.
If a variable y depends on a variable x in a such a way that each
value of x determines exactly one value of y, then we say that
y is a function of x
Def 1.1.2.
A function f is a rule that associates a unique output with each
input rangedomain
A function ( )
rangedomain
Not a function ()
http://images.google.com.my/imgres?imgurl=http://www.analyzemath.com/calculus/continuity/continuous_1.gif&imgrefurl=http://www.analyzemath.com/calculus/continuity/continuous_functions.html&usg=__hdeWlUUoIXXI71svrUD3lXtBVEs=&h=325&w=423&sz=4&hl=ms&start=4&tbnid=nbH7GGesDcYNwM:&tbnh=97&tbnw=126&prev=/images%3Fq%3Dfunctions%26gbv%3D2%26hl%3Dmshttp://images.google.com.my/imgres?imgurl=http://thumb10.shutterstock.com.edgesuite.net/display_pic_with_logo/1934/1934,1155720118,3/stock-photo-arithmetic-operations-1691873.jpg&imgrefurl=http://www.shutterstock.com/pic-1691873/stock-photo-arithmetic-operations.html&usg=__5862epj1JBXL9eYPrDzh7087zDY=&h=358&w=450&sz=20&hl=ms&start=69&tbnid=GHwV0CJdEnE8kM:&tbnh=101&tbnw=127&prev=/images%3Fq%3Darithmetic%2Boperation%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D54http://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D720http://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D720http://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D7207/30/2019 Week1 Functions
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7
Properties
1. Independent & dependent variables
2. Domain & range
)(xfy
Ind ependent var iable =
domain/ input
Dependent variable
=image/range/output
If the independent and dependent variables are real numbers, we say
that fis a real valued fun ct ion of a real var iable
http://images.google.com.my/imgres?imgurl=http://www.analyzemath.com/calculus/continuity/continuous_1.gif&imgrefurl=http://www.analyzemath.com/calculus/continuity/continuous_functions.html&usg=__hdeWlUUoIXXI71svrUD3lXtBVEs=&h=325&w=423&sz=4&hl=ms&start=4&tbnid=nbH7GGesDcYNwM:&tbnh=97&tbnw=126&prev=/images%3Fq%3Dfunctions%26gbv%3D2%26hl%3Dmshttp://images.google.com.my/imgres?imgurl=http://thumb10.shutterstock.com.edgesuite.net/display_pic_with_logo/1934/1934,1155720118,3/stock-photo-arithmetic-operations-1691873.jpg&imgrefurl=http://www.shutterstock.com/pic-1691873/stock-photo-arithmetic-operations.html&usg=__5862epj1JBXL9eYPrDzh7087zDY=&h=358&w=450&sz=20&hl=ms&start=69&tbnid=GHwV0CJdEnE8kM:&tbnh=101&tbnw=127&prev=/images%3Fq%3Darithmetic%2Boperation%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D54http://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D720http://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D7207/30/2019 Week1 Functions
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8
Properties
Function Domain Range
All possible inputs Set of outputs2xy
2,2 xxy
x
x2
y
y4
So to fully describe a function
you must not only specify the
rule that relates the inputs and
outputs, but you must also
specify the domain
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9
Properties
3. Graphs of function
Graphs can provide useful visual information about a function
The values of x for which f(x)=0 are the coordinates of the pointswhere the graph of f intersects the x-axis; These value of x are calledthe zeros of f, the roots of f(x)=0, or th e x-interc epts o f y=f(x)
http://images.google.com.my/imgres?imgurl=http://www.analyzemath.com/calculus/continuity/continuous_1.gif&imgrefurl=http://www.analyzemath.com/calculus/continuity/continuous_functions.html&usg=__hdeWlUUoIXXI71svrUD3lXtBVEs=&h=325&w=423&sz=4&hl=ms&start=4&tbnid=nbH7GGesDcYNwM:&tbnh=97&tbnw=126&prev=/images%3Fq%3Dfunctions%26gbv%3D2%26hl%3Dmshttp://images.google.com.my/imgres?imgurl=http://thumb10.shutterstock.com.edgesuite.net/display_pic_with_logo/1934/1934,1155720118,3/stock-photo-arithmetic-operations-1691873.jpg&imgrefurl=http://www.shutterstock.com/pic-1691873/stock-photo-arithmetic-operations.html&usg=__5862epj1JBXL9eYPrDzh7087zDY=&h=358&w=450&sz=20&hl=ms&start=69&tbnid=GHwV0CJdEnE8kM:&tbnh=101&tbnw=127&prev=/images%3Fq%3Darithmetic%2Boperation%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D54http://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D720http://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D7207/30/2019 Week1 Functions
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10
Properties
4. The vertical line test
A curve in the xy-plane is the graph of some function f if and
only if no vertical line intersects the curve more than once.
Not a function ()2522 yx
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Operations
Two functions, f and g, can be added, substracted, multiplied, and
divided in a natural way to form a new functions f+g, f-g, fg, f/g.
Def. 1.4.1 Given functions f and g, we define
)()())(( xgxfxgf )()())(( xgxfxgf
)()())(( xgxfxfg
)(/)())(/( xgxfxgf For the function f+g, f-g, and fg, we define the domain to be the
intersection of the domains of f and g, and the function f/g we
define the domain to be the intersection of the domains of f and
g but with the points where g(x)=0 excluded
1. Arithmetic operations
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Operations
2. Composition of function
Is performed by substituting some function for the
independent variable of another function
Eg.: Suppose that
2)( xxf and 1)( xxg
))(( xgf2))(( xg
2)1( x gfDenote as,
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Operations
3. Symmetry
about the y-axis about the x-axis about the origin
If replacing x byxin its equationproduces an
equivalent equation
If replacing y byyin its equationproduces an
equivalent equation
If replacing both x byx and y byy in itsequation produces
an equivalent
equation
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Operations
4. Even & odd function
Even function Odd function
)()( xfxf )()( xfxf
Examples:
x
xxx
cos&
,, 642
xs
xxx
in&
,, 53
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Operations
6. Translation
Operation on
y=f(x)
Add a
positive c to
f(x)
subtract a
positive c
from f(x)
Original eqn.
Geommetric
effect
Shifts up c
units
Shifts down c
units
Example
cxfy )( cxfy )(
2
-2
2xy 22 xy 22 xy
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Graphs(Families of Functions)
1. Family of linescxf )(
A function f whose value are all the same is
called a constant function
The graph of the constant function
is the horizontal linecxf )(
cy
If we vary c, then we obtain a set or family of
horizontal lines
cmxy
http://images.google.com.my/imgres?imgurl=http://www.analyzemath.com/calculus/continuity/continuous_1.gif&imgrefurl=http://www.analyzemath.com/calculus/continuity/continuous_functions.html&usg=__hdeWlUUoIXXI71svrUD3lXtBVEs=&h=325&w=423&sz=4&hl=ms&start=4&tbnid=nbH7GGesDcYNwM:&tbnh=97&tbnw=126&prev=/images%3Fq%3Dfunctions%26gbv%3D2%26hl%3Dmshttp://images.google.com.my/imgres?imgurl=http://www.analyzemath.com/calculus/continuity/continuous_1.gif&imgrefurl=http://www.analyzemath.com/calculus/continuity/continuous_functions.html&usg=__hdeWlUUoIXXI71svrUD3lXtBVEs=&h=325&w=423&sz=4&hl=ms&start=4&tbnid=nbH7GGesDcYNwM:&tbnh=97&tbnw=126&prev=/images%3Fq%3Dfunctions%26gbv%3D2%26hl%3Dmshttp://images.google.com.my/imgres?imgurl=http://thumb10.shutterstock.com.edgesuite.net/display_pic_with_logo/1934/1934,1155720118,3/stock-photo-arithmetic-operations-1691873.jpg&imgrefurl=http://www.shutterstock.com/pic-1691873/stock-photo-arithmetic-operations.html&usg=__5862epj1JBXL9eYPrDzh7087zDY=&h=358&w=450&sz=20&hl=ms&start=69&tbnid=GHwV0CJdEnE8kM:&tbnh=101&tbnw=127&prev=/images%3Fq%3Darithmetic%2Boperation%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D54http://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D720http://images.google.com.my/imgres?imgurl=http://www.analyzemath.com/calculus/continuity/continuous_1.gif&imgrefurl=http://www.analyzemath.com/calculus/continuity/continuous_functions.html&usg=__hdeWlUUoIXXI71svrUD3lXtBVEs=&h=325&w=423&sz=4&hl=ms&start=4&tbnid=nbH7GGesDcYNwM:&tbnh=97&tbnw=126&prev=/images%3Fq%3Dfunctions%26gbv%3D2%26hl%3Dmshttp://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D7207/30/2019 Week1 Functions
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Graphs(Families of Functions)
cmxy Fixed c vary m
Family of lines whose
members all have y-intercept b
Fixed m, vary c
Family of parallel lines whose
members all have slope m
x
y
x
y
http://images.google.com.my/imgres?imgurl=http://www.analyzemath.com/calculus/continuity/continuous_1.gif&imgrefurl=http://www.analyzemath.com/calculus/continuity/continuous_functions.html&usg=__hdeWlUUoIXXI71svrUD3lXtBVEs=&h=325&w=423&sz=4&hl=ms&start=4&tbnid=nbH7GGesDcYNwM:&tbnh=97&tbnw=126&prev=/images%3Fq%3Dfunctions%26gbv%3D2%26hl%3Dmshttp://images.google.com.my/imgres?imgurl=http://www.analyzemath.com/calculus/continuity/continuous_1.gif&imgrefurl=http://www.analyzemath.com/calculus/continuity/continuous_functions.html&usg=__hdeWlUUoIXXI71svrUD3lXtBVEs=&h=325&w=423&sz=4&hl=ms&start=4&tbnid=nbH7GGesDcYNwM:&tbnh=97&tbnw=126&prev=/images%3Fq%3Dfunctions%26gbv%3D2%26hl%3Dmshttp://images.google.com.my/imgres?imgurl=http://thumb10.shutterstock.com.edgesuite.net/display_pic_with_logo/1934/1934,1155720118,3/stock-photo-arithmetic-operations-1691873.jpg&imgrefurl=http://www.shutterstock.com/pic-1691873/stock-photo-arithmetic-operations.html&usg=__5862epj1JBXL9eYPrDzh7087zDY=&h=358&w=450&sz=20&hl=ms&start=69&tbnid=GHwV0CJdEnE8kM:&tbnh=101&tbnw=127&prev=/images%3Fq%3Darithmetic%2Boperation%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D54http://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D720http://images.google.com.my/imgres?imgurl=http://www.analyzemath.com/calculus/continuity/continuous_1.gif&imgrefurl=http://www.analyzemath.com/calculus/continuity/continuous_functions.html&usg=__hdeWlUUoIXXI71svrUD3lXtBVEs=&h=325&w=423&sz=4&hl=ms&start=4&tbnid=nbH7GGesDcYNwM:&tbnh=97&tbnw=126&prev=/images%3Fq%3Dfunctions%26gbv%3D2%26hl%3Dmshttp://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D7207/30/2019 Week1 Functions
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Graphs(Families of Functions)
n
xy 2. Power Function1n
4n
3n2n
5n
http://images.google.com.my/imgres?imgurl=http://www.analyzemath.com/calculus/continuity/continuous_1.gif&imgrefurl=http://www.analyzemath.com/calculus/continuity/continuous_functions.html&usg=__hdeWlUUoIXXI71svrUD3lXtBVEs=&h=325&w=423&sz=4&hl=ms&start=4&tbnid=nbH7GGesDcYNwM:&tbnh=97&tbnw=126&prev=/images%3Fq%3Dfunctions%26gbv%3D2%26hl%3Dmshttp://images.google.com.my/imgres?imgurl=http://www.analyzemath.com/calculus/continuity/continuous_1.gif&imgrefurl=http://www.analyzemath.com/calculus/continuity/continuous_functions.html&usg=__hdeWlUUoIXXI71svrUD3lXtBVEs=&h=325&w=423&sz=4&hl=ms&start=4&tbnid=nbH7GGesDcYNwM:&tbnh=97&tbnw=126&prev=/images%3Fq%3Dfunctions%26gbv%3D2%26hl%3Dmshttp://images.google.com.my/imgres?imgurl=http://thumb10.shutterstock.com.edgesuite.net/display_pic_with_logo/1934/1934,1155720118,3/stock-photo-arithmetic-operations-1691873.jpg&imgrefurl=http://www.shutterstock.com/pic-1691873/stock-photo-arithmetic-operations.html&usg=__5862epj1JBXL9eYPrDzh7087zDY=&h=358&w=450&sz=20&hl=ms&start=69&tbnid=GHwV0CJdEnE8kM:&tbnh=101&tbnw=127&prev=/images%3Fq%3Darithmetic%2Boperation%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D54http://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D720http://images.google.com.my/imgres?imgurl=http://www.analyzemath.com/calculus/continuity/continuous_1.gif&imgrefurl=http://www.analyzemath.com/calculus/continuity/continuous_functions.html&usg=__hdeWlUUoIXXI71svrUD3lXtBVEs=&h=325&w=423&sz=4&hl=ms&start=4&tbnid=nbH7GGesDcYNwM:&tbnh=97&tbnw=126&prev=/images%3Fq%3Dfunctions%26gbv%3D2%26hl%3Dmshttp://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D7207/30/2019 Week1 Functions
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Graphs(Families of Functions)
3. Equilateral Hyperbolan
xy
1n 2n
http://images.google.com.my/imgres?imgurl=http://www.analyzemath.com/calculus/continuity/continuous_1.gif&imgrefurl=http://www.analyzemath.com/calculus/continuity/continuous_functions.html&usg=__hdeWlUUoIXXI71svrUD3lXtBVEs=&h=325&w=423&sz=4&hl=ms&start=4&tbnid=nbH7GGesDcYNwM:&tbnh=97&tbnw=126&prev=/images%3Fq%3Dfunctions%26gbv%3D2%26hl%3Dmshttp://images.google.com.my/imgres?imgurl=http://www.analyzemath.com/calculus/continuity/continuous_1.gif&imgrefurl=http://www.analyzemath.com/calculus/continuity/continuous_functions.html&usg=__hdeWlUUoIXXI71svrUD3lXtBVEs=&h=325&w=423&sz=4&hl=ms&start=4&tbnid=nbH7GGesDcYNwM:&tbnh=97&tbnw=126&prev=/images%3Fq%3Dfunctions%26gbv%3D2%26hl%3Dmshttp://images.google.com.my/imgres?imgurl=http://thumb10.shutterstock.com.edgesuite.net/display_pic_with_logo/1934/1934,1155720118,3/stock-photo-arithmetic-operations-1691873.jpg&imgrefurl=http://www.shutterstock.com/pic-1691873/stock-photo-arithmetic-operations.html&usg=__5862epj1JBXL9eYPrDzh7087zDY=&h=358&w=450&sz=20&hl=ms&start=69&tbnid=GHwV0CJdEnE8kM:&tbnh=101&tbnw=127&prev=/images%3Fq%3Darithmetic%2Boperation%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D54http://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D720http://images.google.com.my/imgres?imgurl=http://www.analyzemath.com/calculus/continuity/continuous_1.gif&imgrefurl=http://www.analyzemath.com/calculus/continuity/continuous_functions.html&usg=__hdeWlUUoIXXI71svrUD3lXtBVEs=&h=325&w=423&sz=4&hl=ms&start=4&tbnid=nbH7GGesDcYNwM:&tbnh=97&tbnw=126&prev=/images%3Fq%3Dfunctions%26gbv%3D2%26hl%3Dmshttp://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D7207/30/2019 Week1 Functions
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Graphs(Families of Functions)
n
xy
1
4.
2n
http://images.google.com.my/imgres?imgurl=http://www.analyzemath.com/calculus/continuity/continuous_1.gif&imgrefurl=http://www.analyzemath.com/calculus/continuity/continuous_functions.html&usg=__hdeWlUUoIXXI71svrUD3lXtBVEs=&h=325&w=423&sz=4&hl=ms&start=4&tbnid=nbH7GGesDcYNwM:&tbnh=97&tbnw=126&prev=/images%3Fq%3Dfunctions%26gbv%3D2%26hl%3Dmshttp://images.google.com.my/imgres?imgurl=http://www.analyzemath.com/calculus/continuity/continuous_1.gif&imgrefurl=http://www.analyzemath.com/calculus/continuity/continuous_functions.html&usg=__hdeWlUUoIXXI71svrUD3lXtBVEs=&h=325&w=423&sz=4&hl=ms&start=4&tbnid=nbH7GGesDcYNwM:&tbnh=97&tbnw=126&prev=/images%3Fq%3Dfunctions%26gbv%3D2%26hl%3Dmshttp://images.google.com.my/imgres?imgurl=http://thumb10.shutterstock.com.edgesuite.net/display_pic_with_logo/1934/1934,1155720118,3/stock-photo-arithmetic-operations-1691873.jpg&imgrefurl=http://www.shutterstock.com/pic-1691873/stock-photo-arithmetic-operations.html&usg=__5862epj1JBXL9eYPrDzh7087zDY=&h=358&w=450&sz=20&hl=ms&start=69&tbnid=GHwV0CJdEnE8kM:&tbnh=101&tbnw=127&prev=/images%3Fq%3Darithmetic%2Boperation%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D54http://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D720http://images.google.com.my/imgres?imgurl=http://www.analyzemath.com/calculus/continuity/continuous_1.gif&imgrefurl=http://www.analyzemath.com/calculus/continuity/continuous_functions.html&usg=__hdeWlUUoIXXI71svrUD3lXtBVEs=&h=325&w=423&sz=4&hl=ms&start=4&tbnid=nbH7GGesDcYNwM:&tbnh=97&tbnw=126&prev=/images%3Fq%3Dfunctions%26gbv%3D2%26hl%3Dmshttp://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D7207/30/2019 Week1 Functions
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Graphs(Families of Functions)
BxAy sin BxAy cos5. &
xy sin xy sin2
http://images.google.com.my/imgres?imgurl=http://www.analyzemath.com/calculus/continuity/continuous_1.gif&imgrefurl=http://www.analyzemath.com/calculus/continuity/continuous_functions.html&usg=__hdeWlUUoIXXI71svrUD3lXtBVEs=&h=325&w=423&sz=4&hl=ms&start=4&tbnid=nbH7GGesDcYNwM:&tbnh=97&tbnw=126&prev=/images%3Fq%3Dfunctions%26gbv%3D2%26hl%3Dmshttp://images.google.com.my/imgres?imgurl=http://www.analyzemath.com/calculus/continuity/continuous_1.gif&imgrefurl=http://www.analyzemath.com/calculus/continuity/continuous_functions.html&usg=__hdeWlUUoIXXI71svrUD3lXtBVEs=&h=325&w=423&sz=4&hl=ms&start=4&tbnid=nbH7GGesDcYNwM:&tbnh=97&tbnw=126&prev=/images%3Fq%3Dfunctions%26gbv%3D2%26hl%3Dmshttp://images.google.com.my/imgres?imgurl=http://thumb10.shutterstock.com.edgesuite.net/display_pic_with_logo/1934/1934,1155720118,3/stock-photo-arithmetic-operations-1691873.jpg&imgrefurl=http://www.shutterstock.com/pic-1691873/stock-photo-arithmetic-operations.html&usg=__5862epj1JBXL9eYPrDzh7087zDY=&h=358&w=450&sz=20&hl=ms&start=69&tbnid=GHwV0CJdEnE8kM:&tbnh=101&tbnw=127&prev=/images%3Fq%3Darithmetic%2Boperation%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D54http://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D720http://images.google.com.my/imgres?imgurl=http://www.analyzemath.com/calculus/continuity/continuous_1.gif&imgrefurl=http://www.analyzemath.com/calculus/continuity/continuous_functions.html&usg=__hdeWlUUoIXXI71svrUD3lXtBVEs=&h=325&w=423&sz=4&hl=ms&start=4&tbnid=nbH7GGesDcYNwM:&tbnh=97&tbnw=126&prev=/images%3Fq%3Dfunctions%26gbv%3D2%26hl%3Dmshttp://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D7207/30/2019 Week1 Functions
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Graphs(Families of Functions)
xy cos xy cos2
http://images.google.com.my/imgres?imgurl=http://www.analyzemath.com/calculus/continuity/continuous_1.gif&imgrefurl=http://www.analyzemath.com/calculus/continuity/continuous_functions.html&usg=__hdeWlUUoIXXI71svrUD3lXtBVEs=&h=325&w=423&sz=4&hl=ms&start=4&tbnid=nbH7GGesDcYNwM:&tbnh=97&tbnw=126&prev=/images%3Fq%3Dfunctions%26gbv%3D2%26hl%3Dmshttp://images.google.com.my/imgres?imgurl=http://www.analyzemath.com/calculus/continuity/continuous_1.gif&imgrefurl=http://www.analyzemath.com/calculus/continuity/continuous_functions.html&usg=__hdeWlUUoIXXI71svrUD3lXtBVEs=&h=325&w=423&sz=4&hl=ms&start=4&tbnid=nbH7GGesDcYNwM:&tbnh=97&tbnw=126&prev=/images%3Fq%3Dfunctions%26gbv%3D2%26hl%3Dmshttp://images.google.com.my/imgres?imgurl=http://thumb10.shutterstock.com.edgesuite.net/display_pic_with_logo/1934/1934,1155720118,3/stock-photo-arithmetic-operations-1691873.jpg&imgrefurl=http://www.shutterstock.com/pic-1691873/stock-photo-arithmetic-operations.html&usg=__5862epj1JBXL9eYPrDzh7087zDY=&h=358&w=450&sz=20&hl=ms&start=69&tbnid=GHwV0CJdEnE8kM:&tbnh=101&tbnw=127&prev=/images%3Fq%3Darithmetic%2Boperation%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D54http://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D720http://images.google.com.my/imgres?imgurl=http://www.analyzemath.com/calculus/continuity/continuous_1.gif&imgrefurl=http://www.analyzemath.com/calculus/continuity/continuous_functions.html&usg=__hdeWlUUoIXXI71svrUD3lXtBVEs=&h=325&w=423&sz=4&hl=ms&start=4&tbnid=nbH7GGesDcYNwM:&tbnh=97&tbnw=126&prev=/images%3Fq%3Dfunctions%26gbv%3D2%26hl%3Dmshttp://images.google.com.my/imgres?imgurl=http://asset3.pnn.com/graphics/show/37172/600/image.jpg&imgrefurl=http://rnefian.pnn.com/&usg=___O1DerzGPcpoKGkZWG2e7D2K1OU=&h=500&w=375&sz=139&hl=ms&start=730&tbnid=m9hpNTIwhhRHyM:&tbnh=130&tbnw=98&prev=/images%3Fq%3Dquadratic%2Bfunctions%26gbv%3D2%26ndsp%3D18%26hl%3Dms%26sa%3DN%26start%3D7207/30/2019 Week1 Functions
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Graphs(Families of Functions)
xy 2sin
))2(2sin(
xy
)cos( CBxAy )sin( CBxAy 6.
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Graphs(Families of Functions)
xy 2cos
))2(2cos( xy
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