NEW FUNCTIONS FROM OLD 1. 1.3 New Functions from Old Functions In this section, we will learn: How...
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Transcript of NEW FUNCTIONS FROM OLD 1. 1.3 New Functions from Old Functions In this section, we will learn: How...
NEW FUNCTIONS FROM OLDNEW FUNCTIONS FROM OLD
1
1.3New Functions from Old
FunctionsIn this section, we will learn:
How to obtain new functions from old functions
and how to combine pairs of functions.
FUNCTIONS AND MODELS
Two functions f and g can be combined
to form new functions f + g, f - g, fg, and
in a manner similar to the way we add,
subtract, multiply, and divide real numbers.
COMBINATIONS OF FUNCTIONS
f
g
The sum and difference functions are defined by:
(f + g)x = f(x) + g(x) (f – g)x = f(x) – g(x)
If the domain of f is A and the domain of g is B, then the domain of f + g is the intersection .
This is because both f(x) and g(x) have to be defined.
A B
SUM AND DIFFERENCE
For example, the domain of
is and the domain of
is .
So, the domain of
is .
( )f x x[0, )A ( ) 2g x x
( , 2]B
( ) 2f g x x x [0,2]A B
SUM AND DIFFERENCE
Similarly, the product and quotient
functions are defined by:
The domain of fg is . However, we can’t divide by 0. So, the domain of f/g is
( )( )( ) ( ) ( ) ( )
( )
f f xfg x f x g x x
g g x
A B
| ( ) 0 .x A B g x
PRODUCT AND QUOTIENT
For instance, if f(x) = x2 and g(x) = x - 1,
then the domain of the rational function
is ,
or
2( / )( ) /( 1)f g x x x | 1x x
( ,1) (1, ).
PRODUCT AND QUOTIENT
There is another way of combining two
functions to obtain a new function.
For example, suppose that and
Since y is a function of u and u is, in turn, a function of x, it follows that y is ultimately a function of x.
We compute this by substitution:
( )y f u u 2( ) 1. u g x x
COMBINATIONS
2 2( ) ( ( )) ( 1) 1y f u f g x f x x
This procedure is called composition—
because the new function is composed
of the two given functions f and g.
COMBINATIONS
In general, given any two functions f and g,
we start with a number x in the domain of g
and find its image g(x).
If this number g(x) is in the domain of f, then we can calculate the value of f(g(x)).
The result is a new function h(x) = f(g(x)) obtained by substituting g into f.
It is called the composition (or composite) of f and g. It is denoted by (“f circle g”).
COMPOSITION
f g
Given two functions f and g,
the composite function
(also called the composition of f and g)
is defined by:
f g
( )( ) ( ( ))f g x f g x
DefinitionCOMPOSITION
The domain of is the set of all x
in the domain of g such that g(x) is in
the domain of f. In other words, is defined whenever
both g(x) and f(g(x)) are defined.( )( )f g x
f gCOMPOSITION
The figure shows
how to picture
in terms of machines.
COMPOSITION
f g
If f(x) = x2 and g(x) = x - 3, find
the composite functions and .
We have:
f g g f
2( )( ) ( ( )) ( 3) ( 3)f g x f g x f x x
Example 6COMPOSITION
2 2( )( ) ( ( )) ( ) 3g f x g f x g x x
You can see from Example 6 that,
in general, .
Remember, the notation means that, first,the function g is applied and, then, f is applied.
In Example 6, is the function that first subtracts 3 and then squares; is the function that first squares and then subtracts 3.
f g g f
COMPOSITION Note
f g
g ff g
If and ,
find each function and its domain.
a.
b.
c.
d.
( )f x x ( ) 2g x x
f g
Example 7COMPOSITION
g f
f fg g
The domain of is:
4( )( ) ( ( )) ( 2 ) 2 2f g x f g x f x x x
f g
| 2 0 | 2 ( , 2]x x x x
COMPOSITION Example 7 a
For to be defined, we must have . For to be defined, we must have ,
that is, , or . Thus, we have . So, the domain of is the closed interval [0, 4].
( )( ) ( ( )) ( ) 2g f x g f x g x x
x 0x 2 x 2 0x
2x 4x 0 4x
g f
Example 7 bCOMPOSITION
The domain of is .
COMPOSITION Example 7 c
4( )( ) ( ( )) ( )f f x f f x f x x x
f f [0, )
This expression is defined when both and .
The first inequality means . The second is equivalent to , or ,
or . Thus, , so the domain of is
the closed interval [-2, 2].
( )( ) ( ( ) ( 2 ) 2 2g g x g g x g x x
2 0x 2 2 0x
2x 2 2x 2 4x
2x 2 2x g g
Example 7 dCOMPOSITION
It is possible to take the composition
of three or more functions.
For instance, the composite function is found by first applying h, then g, and then f as follows:
f g h
( )( ) ( ( ( )))f g h x f g h x
COMPOSITION
Find if ,
and .
f g h ( ) /( 1)f x x x 10( )g x x ( ) 3h x x
( f og oh)(x) f (g(h(x)))
f (g(x 3)) f ((x 3)10 ) (x 3)10
(x 3)10 1
Example 8COMPOSITION
So far, we have used composition to build
complicated functions from simpler ones.
However, in calculus, it is often useful
to be able to decompose a complicated
function into simpler ones—as in the
following example.
COMPOSITION
Given , find functions
f, g, and h such that .
Since F(x) = [cos(x + 9)]2, the formula for F states: First add 9, then take the cosine of the result, and finally square.
So, we let:
2( ) cos ( 9)F x x F f g h
2
( ) 9
( ) cos
( )
h x x
g x x
f x x
Example 9COMPOSITION
Then,
2
( )( )
( ( ( )))
( ( 9))
(cos( 9))
[cos( 9)]
( )
f g h x
f g h x
f g x
f x
x
F x
COMPOSITION Example 9