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1.41.41.41.4
Combinations of FunctionsCombinations of Functions
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Quick Review
Find the domain of the function and express it in interval notation.
11. ( )
4
2. ( ) 1
13. ( )
1
4. ( ) log
5. ( ) 4
xf x
x
f x x
f xx
f x x
f x
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What you’ll learn about• Combining Functions Algebraically• Composition of Functions• Relations and Implicitly Defined Functions
… and whyMost of the functions that you will encounter incalculus and in real life can be created by
combining ormodifying other functions.
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The Identity Function
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The Squaring Function
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The Cubing Function
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The Reciprocal Function
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The Square Root Function
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The Exponential Function
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The Natural Logarithm Function
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The Sine Function
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The Cosine Function
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The Absolute Value Function
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The Greatest Integer Function
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The Logistic Function
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Example Looking for Domains
One of the functions has domain the set of all reals except 0.
Which function is it?
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Example Looking for Domains
One of the functions has domain the set of all reals except 0.
Which function is it?
The function 1/ has a vertical asymptote at 0.y x x
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Example Analyzing a Function Graphically
2Graph the function ( -3) . Then answer the following questions.
(a) On what interval is the function increasing?
(b) Is the function even, odd, or neither?
(c) Does the function have any extrema?
y x
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Example Analyzing a Function Graphically
2Graph the function ( -3) . Then answer the following questions.
(a) On what interval is the function increasing?
(b) Is the function even, odd, or neither?
(c) Does the function have any extrema?
y x
(a) The function is increasing on [3, ).
(b) The function is neither even or odd.
(c) The function has a minimum value of 0 at 3.x
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Sum, Difference, Product, and Quotient
Let and be two functions with intersecting domains. Then for all values
of in the intersection, the algebraic combinations of and are defined
by the following rules:
Sum: ( ) ( )
Differ
f g
x f g
f g x f x g x
ence: ( ) ( ) ( )
Product: ( )( ) ( ) ( )
( )Quotient: , provided ( ) 0
( )
In each case, the domain of the new function consists of all numbers that
belong to both the domain of and
f g x f x g x
fg x f x g x
f f xx g x
g g x
f
the domain of . g
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Example Defining New Functions
Algebraically3Let ( ) and ( ) 1. Find formulas of the functions
(a)
(b)
(c)
(d) /
f x x g x x
f g
f g
fg
f g
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Example Defining New Functions Algebraically
3Let ( ) and ( ) 1. Find formulas of the functions
(a)
(b)
(c)
(d) /
f x x g x x
f g
f g
fg
f g
3
3
3
3
(a) ( ) ( ) 1 with domain [ 1, )
(b) ( ) ( ) 1 with domain [ 1, )
(c) ( ) ( ) 1 with domain [ 1, )
( )(d) with domain ( 1, )
( ) 1
f x g x x x
f x g x x x
f x g x x x
f x x
g x x
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Composition of Functions
Let and be two functions such that the domain of intersects the range
of . The composition of , denoted , is defined by the rule
( )( ) ( ( )).
The domain of consists of all -values
f g f
g f g f g
f g x f g x
f g x
in the domain of that
map to ( )-values in the domain of .
g
g x f
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Composition of Functions
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Example Composing Functions
Let ( ) 2 and ( ) 1. Find
(a)
(b)
xf x g x x
f g x
g f x
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Example Composing Functions
Let ( ) 2 and ( ) 1. Find
(a)
(b)
xf x g x x
f g x
g f x
1(a) ( ( )) 2
(b) ( ( )) 2 1
x
x
f g x f g x
g f x g f x
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Example Decomposing Functions
2
Find and such that ( ) ( ( )).
( ) 5
f g h x f g x
h x x
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Example Decomposing
Functions 2
Find and such that ( ) ( ( )).
( ) 5
f g h x f g x
h x x
2
2
One possible decomposition:
( ) and ( ) 5
Another possibility:
( ) 5 and ( )
f x x g x x
f x x g x x
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Example Using Implicitly Defined
Functions2 2Describe the graph of the relation 2 4.x xy y
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Example Using Implicitly Defined
Functions
2 2Describe the graph of the relation 2 4.x xy y
2 2
2
2 4
( ) 4 factor the left side
2 take the square root of both sides
2 solve for
The graph consists of two lines 2 and 2.
x xy y
x y
x y
y x y
y x y x