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College Algebra
Tutorial 30B: Operations with Functions


 

deskLearning Objectives


After completing this tutorial, you should be able to:
  1. Find the sum of functions.
  2. Find the difference of functions.
  3. Find the product of functions.
  4. Find the quotient of functions.
  5. Find the composition of functions.




deskIntroduction



In this tutorial we will be working with functions.  Note as you are going through this lesson that a lot of the things we are doing we have done before with expressions.  Like adding, subtracting, multiplying and dividing.  What is new here is we are specifically looking at these same operations with functions this time.  I think we are ready to forge ahead. 

 

 

desk Tutorial


 
 

The following show us how to perform the different operations on functions.
 

Use the functions function f   and   function g   to illustrate the operations:


 
 
Sum of f + g

(f + g)(x) = f(x) + g(x)

 
This is a very straight forward process.  When you want the sum of your functions you simply add the two functions together.

notebookExample 1: If  function f and   function g  then find (f + g)(x)


 
example 1

*Add the 2 functions 
 

*Combine like terms


 
 
Difference of f - g

(f - g)(x) = f(x) - g(x)

 
 
Another straight forward idea, when you want the difference of your functions you simply take the first function minus the second function.

notebookExample 2: If   function f  and   function g   then find  (f - g)(x) and (f - g)(5)


 
exanoke 2

*Take the difference of the 2 functions
*Subtract EVERY term of the 2nd (  )
 
 
 

*Plug 5 in for x in the diff. of the 2 functions found above
 

 


 
Since the difference function had already been found, we didn't have to take the difference of the two functions again.  We could just merely plug in 5 into the already found difference function.

 
 
Product of f g

(f g)(x) = f(x)g(x)

 
 
Along the same idea as adding and subtracting, when you want to find the product of your functions you multiply the functions together.

notebookExample 3: If   function f  and   function g   then find (fg)(x)


 
example 3

*Take the product of the 2 functions

*FOIL method to multiply
 


 
 
Quotient of f/ g

(f /g)(x) = f(x)/g(x)

 
Well, we don't want to leave division of functions out of the loop.  It stands to reason that when you want to find the quotient of your functions you divide the functions.

notebookExample 4: If   function f     and   function g  then find (f/g)(x) and (f/g)(1)


 
example 4

 

*Write as a quotient of the 2 functions
 
 
 
 
 
 

*Use the quotient found above to plug 1 in for x
 

 


 
 
 
Composite Function

composite function

 
Be careful, when you have a composite function,  one function is inside of the other.  It is not the same as taking the product of those functions.

notebookExample 5: If   function f   and   function g    then find   example 5a


 
example 5b

*g is inside of f

*Substitute in x + 2 for g
*Plug x + 2 in for x in function f
 

 





notebookExample 6:   Let function f,   function g,  and  function h.  Find an equation defining each function and state the domain.

a)  exampl 6a1    b) example 6b1   c) example 6c1   d) exanple 6d1   e)   example 6e1  .




a)  example 6a1


example 6a2
*Add the 2 functions 

 



Domain:

function f

The radicand CANNOT be negative.  In other words it has to be greater than or equal to zero:


example 6a3
*Set radicand of x + 1 greater than or equal to  0 and solve

 



function g

The denominator CANNOT equal zero:


example 6a4
*The den. x CANNOT equal zero


Putting these two sets together we get the domain:

example 6a5


b) example 6b1



example 6b2
*Subtract the 2 functions 

 



Domain:

function f

The radicand CANNOT be negative.  In other words it has to be greater than or equal to zero:


example 6a3
*Set radicand of x + 1 greater than or equal to  0 and solve

 



function g

The denominator CANNOT equal zero:


example 6a4
*The den. x CANNOT equal zero


Putting these two sets together we get the domain:

example 6a5


c) example 6c1


example 6c2
*Multiply the 2 functions 

 



Domain:

function f

The radicand CANNOT be negative.  In other words it has to be greater than or equal to zero:


example 6a3
*Set radicand of x + 1 greater than or equal to  0 and solve

 



function g

The denominator CANNOT equal zero:


example 6a4
*The den. x CANNOT equal zero


Putting these two sets together we get the domain:

example 6a5


d) example 6d1


example 6d2
*Divide the 2 functions 

 



Domain:

function f

The radicand CANNOT be negative.  In other words it has to be greater than or equal to zero:


example 6a3
*Set radicand of x + 1 greater than or equal to  0 and solve

 



function h

The denominator h(x) CANNOT equal zero:


example 6d3
*The den. h(x) = x + 3 CANNOT equal zero


Putting these two sets together we get the domain:

example 6d4



e)   example 6e1


example 6e2

*h is inside of f  is inside of g
*Substitute in x + 3 for h
*Substitute in sqroot(x + 3 + 1) for f(x + 3)


*Plug sqroot(x + 4) in for x in function g



Domain:

The radicand CANNOT be negative AND the denominator CANNOT equal zero.  In other words it has to be greater than zero:


example 6e3
*Set radicand of x + 4 greater than 0 and solve

 



The domain would be:

example 6e4




notebookExample 7:   Let f = {(-3, 2), (-2, 4), (-1, 6), (0, 8)}, and g = {(-2, 5), (0, 7), (2, 9)} and h = {(-3, 0), (-2, 1)}.  Find the following functions and state the domain.

a)  example 6a1    b) example 6b1   c) example 6c1   d) example 6d1   e)   example 7e1




a)  example 6a1


When you are looking for the domain of the sum of two functions that are given as sets, you are looking for the intersection of their domains.

Since the x values that f and g have in common are -2 and 0, then the domain would be {-2, 0}.



x = -2:

example 7a2

(-2, 9)

*Add together the corresponding y values to x = -2



x = 0:

example 7a3

(0, 15)

*Add together the corresponding y values to x = 0



Putting it together in ordered pairs we get:

example 7a4



b) example 6b1


When you are looking for the domain of the difference of two functions that are given as sets, you are looking for the intersection of their domains.

Since the x values that f and g have in common are -2 and 0, then the domain would be {-2, 0}.



x = -2:

example 7b2

(-2, -1)

*Subtract the corresponding y values to x = -2



x = 0:

example 7b3

(0, 1)

*Subtract the corresponding y values to x = 0



Putting it together in ordered pairs we get:

example 7b4



c) example 6c1


When you are looking for the domain of the product of two functions that are given as sets, you are looking for the intersection of their domains.

Since the x values that f and g have in common are -2 and 0, then the domain would be {-2, 0}.



x = -2:

example 7c2

(-2, 20)

*Multiply the corresponding y values to x = -2



x = 0:

example 7c3

(0, 56)

*Multiply the corresponding y values to x = 0



Putting it together in ordered pairs we get:

example 7c4




d) example 6d1


When you are looking for the domain of the quotient of two functions that are given as sets, you are looking for the intersection of their domains AND values of x that do NOT cause the denominator to equal 0.

The x values that f and h have in common are -3 and -2.  However, h(-3) = 0, which would cause the denominator of the quotient to be 0.

So, the domain would be {-2}.



x = -2:

example 7d3

(-2, 4)

*Find the quotient of the corresponding y values to x = -2



Putting it together in an ordered pair we get:

example 7d4



e)   example 7e1


When you are looking for the domain of the composition of two functions that are given as sets, you are looking for values that come from the domain of the inside function AND when you plug those values of x into the inside function, the output is in the domain of the outside function.

The x values of h are -3 and -2.  However, g(h(-2)) = g(1), which is undefined.  In other words, there are no ordered pairs in g that have a 1 for their x value.


So, the domain would be {-3}.



x = -3:

example 7e2

(-3, 7)
*h(-3) = 0

*Find the y value that corresponds to x = 0



Putting it together in an ordered pair we get:

example 7e4



notebookExample 8:  Let function fand  function g.  Write  function h as a composition function using f and g.




When you are writing a composition function keep in mind that one function is inside of the other.  You just have to figure out which function is the inside function and which is the outside function.

Note that if you put g inside of f you would get:



example 8a

*Put g inside of f



Note that if you put f inside of g you would get:


example 8b

*Put f inside of g



Hey this looks familiar.

Our answer is   example 8c.



 

desk Practice Problems



These are practice problems to help bring you to the next level.  It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it.  Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument.  In fact there is no such thing as too much practice.

To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that  problem.  At the link you will find the answer as well as any steps that went into finding that answer.

 

pencilPractice Problems 1a - 1i: If problem 1 function f   and   problem 1 function g then find the following:

 

 

 

 

 

 

pencilPractice Problems 2a - 2e: If f = {(1, 2), (2, 3), (3, 4), (4, 5)}, g = {(1, -2), (3, -3), (5, -5)}, and h = {(1, 0), (2, 1), (3, 2)}, find the following and state the domain:


 





 

pencilPractice Problem 3a: Let function f  and  function gWrite the given function as a composition function using  f and g.


 

 
 

 

desk Need Extra Help on these Topics?



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Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.


 

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Last revised on March 31, 2010 by Kim Seward.
All contents copyright (C) 2002 - 2010, WTAMU and Kim Seward. All rights reserved.