mth 104 calculus and analytical geometry lecture no. 2
TRANSCRIPT
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MTH 104Calculus and Analytical Geometry
Lecture No. 2
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Functions
If a variable depends on a variable x in such a way that each value of determines exactly one value of , then we say that is a function of . For example ,
y
x
y y
1xyx
x y
-1 -2
0 -1
1 0
2 1
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Functions can be represented in four ways.• Numerically by tables• Geometrically by graphs• Algebraically by formulas• Verbally
Functions
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Functions
• A function is a rule that associates a unique output with each input. If the input is denoted by , then the output is denoted by . Sometimes we will want to denote the output by a single letter, say , and write
f
x).(xf
y
)(xfy
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Graphs of functions
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Graphs of functions
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Graphs of functions
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Graphs of functions
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Graphs of functions
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Functions: Vertical line test
• The vertical line test. A curve in the xy-plane is the graph of some function if and only if no vertical line intersects the curve more than once.
Consider the following four graphs:
f
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Functions: Vertical line test
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The absolute value function
• The absolute value or magnitude of a real number is defined by
x
0 x,
0 ,
x
xxx
Illustration: 5 3
00
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Properties of absolute values
If a and b are real numbers, then
(i) (ii)
(iii)
(iv)
aa
baab
0 , bb
a
b
a
baba
Functions defined piecewise. The absolute value function is defined piecewise
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Functions: Domain and Range
If and are related by the equation , then the set of all allowable inputs is called the domain of , and the set of outputs that results when
varies over the domain is called the range of . Natural domain:
If a real-valued function of a real variable is defined by a formula, and if no domain is stated explicitly, then it is to be understood the domain consists of all real numbers for which the formula yields a real value. This is called the natural
domain of the function
x y )(xfy )values( x
f )values( y x
f
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Functions: Domain and Range
Example. Find the natural domain of
3)( )( xxfa )3)(1(
1)( )(
xxxfb
xxfc tan)( )( 65)( )( 2 xxxfd
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Arithmetic operations on functions Given functions and , we define
For the functions and we define the domain to be the intersection of the domains of and , and for the function we define the domain to be the intersection of the domains of and but with the points where excluded ( to avoid division by zero).
f g
)()())(( xgxfxgf
)()())(( xgxfxgf )()())(( xgxfxfg )(/)())(/( xgxfxgf
gfgf , fg
f g gf /
fg 0)( xg
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Arithmetic operations on functionsExample 1: Let and . Find the domains
and formulas for the functions
Example 2: Show that if , and , then the
domain of is not the same as the natural domain of . • Example 3: Let . Find
• (a) (b) (c)
21)( xxf 3)( xxg
.7 and /,,, fgffggfgf xxf )( xxg )( xxh )(
fg h
1)( 2 xxf
)( 2tf
x
f1 )( hxf
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Composition of functions Given functions and , the composition of with , denoted by ,
is defined by
The domain of is defined to consist of all in the domain of for which is in the domain of
Example Let and . Find
(a) (b) and state the domains of the
compositions.
f g gf gf
))(()( xgfxgf gf x
g )(xg .f
3)( 2 xxf xxg )(
)(xgf )(xfg
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Composition of functions
Composition can be defined for three or more functions: for example is computed as
Example Find if
))(( xhgf
)))((()( xhgfxhgf
)(xhgf 3)( ,1
)( ,)( xxhx
xgxxf
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Expressing a function as a composition
ConsiderLet then in terms of can be
written as
Example Express as a composition of two
functions.
2)1()( xxh
,1)( xxg2)( xxf h gf and
5)4()( xxh