Composition of functions

If f: A → B, g: B → C

Then

gof : A → C

gof = g(f(x))

Here, gof is formed by the composition of functions f and g.

In gof :

• Value of x is coming from set A
• Value of function gof will be from set C

Let us take an example

Let f: {1, 2, 3, 4} → {5, 6, 7, 8}

f(1) = 5, f(2) = 6, f(3) = 7, f(4) = 8

and

g: {5, 6, 7, 8} → {9, 10, 11, 12}

g(5) = 9, g(6) = 10, g(7) = 11, g(8) = 12

Find gof

gof will be

gof (1) = 10

gof (2) = 11

gof (3) = 12

gof (4) = 13

Let’s take another example

f: R → R , g: R → R

f(x) = sin x , g(x) = x ³

Find fog and gof

f(x) = sin x

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

f og (x) = sin (x ³ )

g(x) = x ³

g(f(x)) = f(x) ³

go f (x) = sin ³ x

Note that go f ≠ f og .