Operations on functions

This lesson will teach you how to perform operations on functions. Basically, you can add, subtract, multiply, and divide functions. Let f and g be functions. Take a look at the following figure to see how we can perform these operations on a function.

More examples showing how to do operations on functions.

How to do addition of functions

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

Example #1

Find f + g if  f(x) = 2x - 7 and g(x) = 5x + 8

(f + g)(x)   =   f(x) + g(x)

                 = 2x - 7 + 5x + 8

                 = 2x + 5x + - 7 + 8
 
                = 7x + 1

How to do subtraction of functions

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

Example #2

Find f - g if  f(x) = 3x + 9 and g(x) = 6x - 3

(f - g)(x)   =   f(x) - g(x)

                 = 3x + 9  -  (6x - 3)

                 = 3x + 9  - 6x - - 3
  
                 = 3x + 9 - 6x + 3

                 = 3x - 6x + 9 + 3

                = -3x + 12

How to do multiplication of functions

( f × g )(x) = f(x) × g(x)

Example #3

Find f × g if f(x) = x + 1 and g(x) = x + 5

(f × g)(x) = f(x) × g(x)

= (x + 1) × (x + 5)

= x × x + x × 5 + 1 × x + 1 × 5

= x² + 5x + 1x + 5

x² + 6x + 5

How to do division of functions

( f / g )(x) = f(x) / g(x)     with g(x) ≠ 0

Example #4

Find f / g if f(x) = x² - 1 and g(x) = x + 1

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

= (x² - 1) / (x + 1)    with x + 1 ≠ 0

= [(x - 1) × (x + 1)] / x + 1    with x + 1 ≠ 0

= x - 1     with x + 1 ≠ 0

= x - 1     with x ≠ -1