1.41.4 combinations of functions. quick review what you’ll learn about combining functions...
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1.41.41.41.4
Combinations of FunctionsCombinations of Functions
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
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.
The Identity Function
The Squaring Function
The Cubing Function
The Reciprocal Function
The Square Root Function
The Exponential Function
The Natural Logarithm Function
The Sine Function
The Cosine Function
The Absolute Value Function
The Greatest Integer Function
The Logistic Function
Example Looking for Domains
One of the functions has domain the set of all reals except 0.
Which function is it?
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
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
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
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
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
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
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
Composition of Functions
Example Composing Functions
Let ( ) 2 and ( ) 1. Find
(a)
(b)
xf x g x x
f g x
g f x
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
Example Decomposing Functions
2
Find and such that ( ) ( ( )).
( ) 5
f g h x f g x
h x x
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
Example Using Implicitly Defined
Functions2 2Describe the graph of the relation 2 4.x xy y
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
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