Arithmetic Combination of Functions:

Just how numbers can be added, subtracted, multiplied, and divided --so can functions. Two functions can be combined to create new functions.

The domain of an arithmetic compound  of functions f and g consists of all real numbers.  In the case of f(x)/g(x), g(x) cannot = 0. 

The picture below describes the sum, difference, product, and quotient of functions:

Examples:

Finding the Sum of Two Functions:

Find (f+g)(x) for the functions: f(x)= x+1   and g(x)= (x-1). Then evaluate the sum when x=3

(f+g)(x)= f(x)+ g(x)                   Break up the equation

               (x+1)   +  (x-1)           Plug in the values for f(x) and g(x)

               x+x+1-1                      Add

              = 2x                             Final answer

Solve when x=3:   

(f+g)(3)= 2(3)  = 6

Finding the Difference of Two Functions:

Find (f-g)(x) for the functions: f(x)= (x+1)   and g(x)= (x-1). Then evaluate the sum when x=3

(f-g)(x)= f(x) - g(x)                   Break up the equation

               (x+1)   -  (x-1)           Plug in the values for f(x) and g(x)

               x- x+1+1                      Add

              = 2                             Final answer

Solve when x=3:   

(f-g)(3)= (3+1)  - (3-1) 

               4-3+1  

               1+1= 2

Finding the Product of Two Functions:

Find (f+g)(x) for the functions: f(x)= x+1   and g(x)= (x-1). Then evaluate the sum when x=3

(f*g)(x)= f(x) * g(x)                   Break up the equation

               (x+1)  *  (x-1)           Plug in the values for f(x) and g(x)

               x² +x-x-1                  Foil

              = x² -1                      Final Answer
                  

              

Solve when x=3:   

(f*g)(3)=   x² -1    

                  (3)2  -1   

                   9-1

                   =8

Finding the Quotient of Two Functions:

Find (f/g)(x)  for the functions: f(x)= x+1   and g(x)= (x-1). Then evaluate the sum when x=3

(f/g)(x)= f(x) / g(x)                   Break up the equation

               (x+1) / (x-1)           Plug in the values for f(x) and g(x)

               = (x+1)/(x-1)            Final Answer

The domain of this function is all real numbers except x=1  (since the denominator cannot equal zero).

Written in interval notation, the domain is: (-∞, 1) U (1, ∞)

For more helpful information regarding the combination of functions, see the following videos:

 

By: Allison Davis, 9/25/12