```Combinations of
Functions:
Composite
Functions
Section 2.6
College Algebra, MATH 171
Mr. Keltner
“Combining”
Functions

If we have two functions, call them f(x) and g(x),
then we can combine them to form new functions:





(f + g)(x)
(f - g)(x)
(f  g)(x)
(f/g)(x)
This is done very similar to how we combine real
numbers, or combine like terms (terms with the
same variables and exponents, such as -5x2 and
8x2) in a long expression.
Combining
Functions:
What it looks like

We can define the function (f + g)(x) as
simple as combining the expressions for
f(x) and g(x), or like this:



(f + g)(x) = f(x) + g(x)
The new function, f + g, is called the
sum of the functions f and g.
The domain of our new function is all
the x-values that were in the domain of
both f(x) and g(x).
Combining Functions:
A little name-calling

Similar to defining the function (f + g)(x) as the sum
of the functions f(x) and g(x), we can also define
these other combinations of functions f(x) and g(x):
(f + g)(x) = f(x) + g(x)
(f - g)(x) = f(x) - g(x)
(f  g)(x) = f(x) g(x)
(f/g)(x) = f(x) ÷ g(x), where g(x) ≠ 0

With the quotient, you must also remember that we
do not want to divide by zero, and determine the
values of the domain accordingly.
Example 1

Assume f x  5x 3 and gx  11x

Find the following:
1

f(x) + g(x)

 f(x) - g(x)


Evaluate each when x = 2.
1
3
Example 2


Let f x   8x and gx  2x
Find the following:





5
6
f(x)  g(x)
f x 
g x 

Find the domain of each new function.
Composition of
Functions


When we show that one function’s result
depends on another function’s result, we
can describe this by working with a
composition of functions.
If we say that h(x) is a composition of
functions f(x) and g(x), we can write this
as: hx  f gx

It is also sometimes written as (f  g)(x).
Composition of
Functions:
Importance of Order

Just because f(g(x)) looks like multiplication
does not mean we can apply properties of
multiplication when working with them.




In other words, there are only certain special
instances where f(g(x)) = g(f(x)).
This is discussed in section 2.7 in particular and
points out a unique feature of the functions f(x) and
g(x).
So, in general, we conclude that f(g(x)) ≠ g(f(x)).
The next example illustrates why this is true.
Composition of
Functions
“Practical” Example


Kohl’s sends us a BUNCH of coupons as a “Most Valuable
Customer” promotion. (I think we’re the most awesomest,
but I don’t teach English, obviously)



If there is no fine print saying we can’t
use BOTH coupons, in which order
should we ask the clerk to ring them up?



\$10 off any purchase
20% off any one day’s purchases
\$10 off first, or
20% off first?
Write each scenario as an equation to
see what you still pay for.

Try a \$45 purchase for example.
Steps in Composition
of Functions


Example 3:
Let f(x) = -3x - 4 and
g(x) = x2 - 1.
Find the value of f(g(-3)).


Evaluate g(-3) by inserting
-3 into g(x) and
simplifying to ______.
Now, use that value in f(x)
to finish up. This means
we are evaluating
f(_____).
1)
2)
3)
Evaluate (or substitute
in) the function in the
inner-most
parentheses first.
Using function notation,
substitute this value (or
expression) wherever
you see a variable in
the function outside the
parentheses.
Simplify, when
applicable.
Example 4


Let f(x) = 6x-2 and g(x) = 4x + 5.
Find each of the following:

f(g(x))

g(f(x))

g(g(x))

Example 5: Composition
of
3 functions

Find f  g  h given the following
information about f(x), g(x), and h(x):
x
f x 
x 1

gx  x10

h x   x  3
Recognizing a
Composition of
Functions

Given the function F x  x 2  2x  3 ,
find functions f(x) and g(x) such that
F(x) = f g.

For a composition
of functions, remember
f g = f(g(x)).
So, if we can figure out which
quantity we need to calculate first,
we also know our inside function, g(x).
Assessment
Pgs. 238-240:
#’s 3 - 36, 45 - 63, multiples of
3
```