Learn Operations on Functions

Generally a function is represented in terms of  f(x).

Functions behave exactly as one would expect with regard to the four basic operations of algebra (addition, subtraction, multiplication, and division). When functions are combined by these operations, though, the domain of the new combined function is only the elements that were shared by the domains of the original functions.

Operations on functions :

As we add, subtract, multiply the numbers, we can add , subtract, multiply, divide the functions. There is one operation more called the composition of functions.

Let us learn through examples, that would be easier to understand.

Consider the two functions,

f(x) = x +2

g (x) = x2 -5


Operations on functions : Addition


f(x) = x +2

g (x) = x^2 -5

(f+g) (x) = x +2 +x2 -5

= x2+x -5 +2

= x2 + x -3

(f+g) (2) = 22 +2 -3 = 6 -3 =3


Operations on functions : Subtraction


f(x) = x +2

g (x) = x^2 -5

(f-g) (x) = x+2 – (x2 -5)

= x2+x +2 -5

=x2 +x -3

(f-g)(2) = 2 +2 - 22 -5 = -5


Operations on functions : Multiplication


f(x) = x +2

g (x) = x^2 -5

(f)(g)(x) = (x+2) (x2-5)

= x3 -5x +2x2 -10

= x3 +2x2 -5x -10

(f)(g)(2) = (2+2) (22-5) = -4


Operations on functions : Division


f(x) = x +2

g (x) = x2 -5

g(x) /f(x) =

=  [x2 -5] / [x+2]

g (2) / f (2 ) = (22 - 5)/ (2+2)

=(4 -5)/ (2+2) = -1 / 4


Operations on functions : Composition of Function


f(x) = x +2

g (x) = x^2 -5

(f o g)(x) read as f of g of x, we can say  that first, we find g(x) and then we find f(x) for the result that we got from g(x)

The concept is simple. First, the value of g at x is taken, and then the value of f at that value is taken

g(x) = x2 -5

(f o g)(x)= f (x2-5) = x2 -5 +2 = x2 -3

(f o g)(2) =

= g(2) =22 – 5 = -1

= f(-1) = -1+2 =1