1.4 - Arithmetic Combinations of Functions

Just as two real numbers can be combined by the operations of addition, subtraction, multiplication, and division to form other real numbers, two functions can be combined to create new functions. 

Now, instead of solving problems like 10+12, 4 x 6, or 8-2, you will be solving problems in which functions are undergoing operations. 


Sum, Difference, Product, and Quotient of Functions:

Just like basic integers you have been used to working with, functions have operations as well. These are the operations you will be dealing with.

Let f and g be two functions with overlapping domains.

1. Sum:                          (f+g)(x) = f(x) + g(x)

2. Difference:                 (f-g)(x) = f(x) - g(x)

3. Product:                      (fg)(x) = f(x) ⋅ g(x)

4. Quotient:                     (f/g)(x) = f(x)/g(x),    g(x) ≠ 0

Finding the Sum of Two Functions

To find the sum of two functions, simply use your knowledge of combining like terms when dealing with polynomials to solve. Add just as you would basic integers. When asked to evaluate the system, simply plug in the number provided to come to a solution. 

Example:


Find (f + g)(x) for the functions f(x) = 2x + 1 and g(x) = x^2 + 2x - 1. Then evaluate the system when x=2.

(f + g)(x) = f(x) + g(x) 
                = (2x + 1) + (x^2 + 2x - 1)
                = x^2 + 4x

When x = 2, the value of this sum is (f+g)(2) = 2^2 + 4(2) = 12




Finding the Difference of Two Functions

To find the difference of two functions, you perform the same steps as when finding the sum of two functions, except this time you will be finding (f-g)(x) instead of (f+g)(x). 

Example:

Evaluate (f - g)(x) for the functions 

f(x) = 2x+1 and g(x) = x^2 + 2x - 1 when x=2

The difference of the functions f  and g is

(f - g)(x) = f(x) - g(x) 
              = (2x+1) - (x^2+2x-1)
              = -x^2 + 2

When x = 2, the value of this difference is 

(f - g)(2) = -(2)^2 + 2
               = -2


Finding the Product of Two Functions

We are now moving into the multiplication of functions. The process is the same as addition and subtraction, except that it is (fg)(x) instead of (f + g)(x) or (f - g)(x). 

Example:

Given that the functions f(x) = x^2 and g(x) = x - 3, find the product of f and g. 
Then evaluate the product when x = 4. 

(fg)(x) = f(x)g(x)
           =(x^2)(x-3)
           = x^3 - 3x^2. 

When x = 4, the value of this product is 

(fg)(4) = 4^3 - 3(4)^2
           = 16.

Finding the Quotient of Two Functions

Finally, we come to the last operation, division. Just like all the others, the process is the same. The thing that sets finding the quotient apart is that we now need to take the domains of f/g and g/f. 

Example:

Find (f/g)(x) and (g/f)(x) for the functions f(x) = √x and g(x) = √4-x^2.
Then find the domains of f/g and g/f.

The quotient of f and g is:

(f/g)(x) = f(x) / g(x) =√x / √4 - x^2

and the quotient of g and f  is:

(g/f)(x) = g(x) / f(x) = √4 - x^2 / √x


The domain of f  is [0,∞) and the domain of g is [-2, 2]. The intersection of these domains in [0, 2]. So the domains for f/g and g/f  are as follows. 

Domain of f/g: [0, 2)        Domain of g/f: (0, 2]