Chapter 1: Linear
relations and functions
Section 1-2
Composition of Functions
Objectives
„
„
„
Perform operations with functions
Find composite functions
Iterate functions using real numbers
Operations with Functions
„
„
„
„
Sum: (f + g )(x) = f (x)+ g (x)
Difference: (f - g) (x) = f (x) – g (x)
Product: ( f ⋅ g ) ( x ) = f (x) ⋅g (x)
Quotient:
⎛ f ⎞
⎜ ⎟ x = f ( x) , g ( x) ≠ 0
⎜ g ⎟
g ( x)
⎝ ⎠
( )
Example #1
„
Given f (x) = 2x – 1 and g (x) = x 2 , find
each function.
( f + g ) (x)
( f – g ) (x)
( f ⋅ g )(x )
⎛f⎞
⎜⎜ ⎟⎟(x)
⎝g⎠
Solution
(f + g) (x) = f (x) + g (x)
So it is 2x – 1 +
x
2
(f – g) (x) = f (x) – g (x)
So it is 2x – 1 – ( x 2 ) =
− x2 + 2 x − 1
Solutions
(f ⋅g)(x)=
f ( x) ⋅ g ( x)
So (2x-1)( x 2) = 2 x3 − x 2
⎛f⎞
f ( x)
⎜⎜ ⎟⎟( x) =
g ( x)
⎝g⎠
2x −1
x≠0
2
x
Composite Functions
The function formed by composing two or more
functions is called the composite. It is denoted
by o . For example, composing two ofunctions f
and g would be written f o g and would be
read as “f composition g” or f of g.
IMPORTANT: f of g is NOT the same as g of f
Composition of Functions
Given functions f and g, the composite function
f o g can be described by the following
equation.
(f o g )(x)=f (g (x))
The domain of f o g includes all of the
elements x in the domain of g for which g (x)
is the domain of f.
Function Machine
Function
X
Y
You chose the x’s that you wish to put in the function machine. It
performs the function and then spits out the answer. For example,
suppose the function was 2x – 1. You put a 3 in the machine multiplies
it by 2 and then subtracts 1, it then spits out the answer which is Y. The
x is the independent variable because you choose it and the Y is the
dependent variable because it depends on what you put in to the
machine. When you have more than one function (composite) you then
have several machines.
Composite of Functions
Suppose we are asked to find (f
and g (x) = 3x
o g )(x)
and (g
o f )(x)
for f (x)=
x2 −1
g (x)
f (x)
3 ( x2
−1
)
x2 −1
x
o
So f
g (x) =
3x − 3
2
g (x)
f (x)
3x
(3 x) 2 − 1
x
Notice these are
not the same
3x 2 − 3
So g
o
f (x) = 9 x − 1
2
9x2 −1
The Domain
State the domain of f o g (x) for f (x) =
and
g (x)
=
x−6
1
x2 − 2
o then that
If g (x) is undefined for a given value of x,
value is excluded from the domain of f g (x) . So
in this example x ≠ ± 2 . The domain of f (x) is
x ≥ 6 . So for x to be in the domain of f g (x), it
must be true that g (x) ≥ 6.
The domain
1
≥6
2
x −2
1 ≥ 6 x 2 − 12
13 ≥ 6 x 2
13
≥ x2
6
13
13
−
≤x≤
, x≠± 2
6
6
Iteration
The composition of a function and itself
(f ( f (x)) is called iteration. Each output of an
iterated function is called an iterate. To iterate
a function f (x), find the function value f ( x0 ),
of the initial value x0 . The value f ( x0 ) is the
first iterate x1 . Continue this process to find
the second and third value.
Remember f ( f ( x0 )= f (
so forth.
x1
), f ( f ( x1 ) = x2 and
Iteration
Find the first three iterates x1 , x2 , x3 of the
function f (x) = 3x + 2 for an initial value of
x0 = 4
To obtain the first iterate, find the value of the
function for
x0 = 4
x1 = f ( x0 ) = f (4) = 3(4) + 2
So x1 = 14
x2 = f ( x1 ) = f (14) = 3(14) + 2 = 44
So x 2 = 44
x3 = f ( x2 0 = f (44) = 3(44) + 2 = 134
So x 3 = 134
HW #2
Section 1-2
PP 17-19
#11, 13, 14, 15, 17, 18, 21, 25, 35, 36, 39