Transcript 4-2 Notes

4-2 Operations on Functions
Just like real numbers, you can add subtract,
multiply, and divide functions to create NEW
functions.
Let f and g be two functions with overlapping
domains. Then for all x common to both
domains:
Sum :
Difference :
Product :
Quotient :
( f  g )( x )  f ( x )  g ( x )
( f  g )( x )  f ( x )  g( x )
( fg )( x )  f ( x )  g ( x )
 f 
f(x)
 ( x ) 
, g( x )  0
g( x )
g
Sum Example
Given : f ( x )  2 x  1 and g( x )  x 2  2 x  1, find ( f  g )( x ).
Then evaluate the sum when x  2.
TI-84?
Difference Example
Given : f ( x )  2 x  1 and g( x )  x 2  2 x  1, find ( f  g )( x ).
Then evaluate the difference when x  2.
TI-84?
Product Example
Given : f ( x )  x 2 and g( x )  x  3, find( fg )( x ).
Then evaluate the product when x  4.
TI-84?
Quotient Example
Given : f ( x )  x and g( x )  4  x 2 , find ( f / g )( x ) and ( g / f )( x ).
Then find the domains of f/g and g/f .
Composition
The compositio n of the function f with the function g is :
( f  g )( x )  f ( g( x )).
The domain of f  g is the set of all x in the domain of g such
that g(x) is in the domain of f .
Composition Example
Given : f ( x )  x , x  0 and g( x )  x  1, x  1, find( f  g )( x ).
If possible, find ( f  g )( 2 ) and ( f  g )( 0 ).
More Composition
Given : f ( x )  x  2 and g( x )  4  x 2 , find ( f  g )( x ) and ( g  f )( x ).
If possible, evaluate each composition at x  0, 1, 2, and 3.
Doublecheck: Finding the Domain
of a Composition
Given : f ( x )  x 2  9 and g( x )  9  x 2 , find ( f  g )( x ) and
state the domain.
Do you remember this special
case? What does it mean?
1
Given : f ( x )  2 x  3 and g( x )  ( x  3 ), find ( f  g )( x ) and
2
.( g  f )( x ).
If f(g(x)) = g(f(x)) = x, then the two functions
are inverses of each other! Graphically, the
functions are symmetric about the line y = x.
Decomposing?
In calculus, it will become important to be able
to identify two functions that make up a given
composite function. Basically, to “decompose” a
composite function, look for an “inner” and an
“outer” function.
Write the function h( x )  ( 3x  5 )3 as a composition of two functions.
h(x) = (3x – 5)3
f(x) = x3
g(x) = 3x – 5
h(x) = f(g(x))
You Try It!
Write the function h( x ) 
1
( x2)
2
as a compositio n of two functions.
Who does this stuff?
The number N of bacteria in a refridgera ted food is given by
N ( T )  20T 2  80T  500 ,
2  T  14
where T is the temperatu re of the food in degrees C. When the
food is removed from refridgera tion, the temperatu re of the food
is given by
T ( t )  4t  2,
0t 3
where t is the time in hours.
Find the compositio n N ( T ( t )) and interpret its meaning in context.
Find the number of the bacteria in the food when t  2 hours.
Find the time when the bacterial count reaches 2000.
Homework
Pages 128-130, #1, 3, 5-10, 17-19, 23-26, 33, 35