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 -5 x 2 and
8 x 2 ) 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.


  
5 x
1
Assume and

Find the following:
 f(x) + g(x) f(x) - g(x)

   
11 x
1
3

Evaluate each when x = 2.



  
8 x

Find the following:
  
2 x
5
6

 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:
  
 

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)
We receive two coupons:

$10 off any purchase

20% off any one day’s purchases
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 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) = -3 x - 4 and g(x) = x 2 - 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) = 6 x -2 and g(x) = 4 x + 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 x

1
   x
10
   x

3
 

Recognizing a
Composition of
Functions

   x
2 
2 x

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).

Pgs. 238-240:
#’s 3 - 36, 45 - 63, multiples of
3