HOW TO FIND COMPOSITION OF TWO FUNCTIONS

About "How to Find Composition of Two Functions"

How to Find Composition of Two Functions

Here we are going to see, how to find composition of two functions.

Definition of Composition of two Functions

Let f : A -> B and g : B -> C be two functions. Then the composition of f and g denoted by g o f is defined as the function g o f (x) = g(f (x)) for all x  A.

Generally, f o g  g o f for any two functions f and g. So, composition of functions is not commutative.

Question 1 :

Using the functions f and g given below, find f o g and g o f . Check whether  f o g = g o f .

(i) f (x) = x −6, g(x) = x2

Solution : 

f o g (x)  =  f[g(x)]

We apply the function given for g(x).

f o g (x)  =  f[x2]

Instead of x, we have x2, so we apply  x2 instead of x in f(x).

f o g (x)  =  f(x2) = x2 − 6   ---(1)

g o f (x)  =  g[f(x)]

=  g [x- 6]

Instead of x, we have x - 6, so we apply  x - 6 instead of x in g(x).

g o f (x)  =  g(x - 6)  =  (x - 6)2  ---(2)

f o g (x)  g o f (x)

(ii) f (x) = 2/x, g(x) = 2x2 - 1 

Solution : 

f o g (x)  =  f[g(x)]

We apply the function given for g(x).

f o g (x)  =  f[2x- 1]

Instead of x, we have 2x- 1, so we apply  2x- 1 instead of x in f(x).

f o g (x)  =  f(2x- 1) = 2/(2x- 1)   ---(1)

g o f (x)  =  g[f(x)]

=  g [2/x]

Instead of x, we have 2/x, so we apply  2/x instead of x in g(x).

g o f (x)  =  g(2/x)  = 2(2/x)2 - 1

=  2(4/x2) - 1

=  (8/x2) - 1   ---(2)

f o g (x)  g o f (x)

(iii) f (x) = (x + 6)/3,  g(x) = 3 - x 

Solution : 

f o g (x)  =  f[g(x)]

We apply the function given for g(x).

f o g (x)  =  f[3 - x]

Instead of x, we have 3 - x, so we apply  3 - x instead of x in f(x).

f(3 - x)  =  (3-x+6)/3

=  (9-x)/3  ---(1)

g o f (x)  =  g[f(x)]

=  g [(x + 6)/3]

Instead of x, we have [(x + 6)/3], so we apply  [(x + 6)/3] instead of x in g(x).

 g [(x + 6)/3]  =  3 - [(x + 6)/3]

  =  (9 - x - 6)/3

=  (3 - x)/3  ---(2)

f o g (x)  g o f (x)

(iv) f (x) = 3 + x,  g(x) = x - 4

Solution : 

f o g (x)  =  f[g(x)]

We apply the function given for g(x).

f o g (x)  =  f[x - 4]

Instead of x, we have x - 4, so we apply x - 4 instead of x in f(x).

f(x - 4)  =  3 + x - 4

=  x - 1  ---(1)

g o f (x)  =  g[f(x)]

=  g [3 + x]

Instead of x, we have 3 + x, so we apply 3 + x instead of x in g(x).

 g [3 + x]  =  3 + x - 4 

  = x - 1  ---(2)

(1)  =  (2)

f o g (x)  =  g o f (x)

(v)  f (x) = 4x2 − 1, g(x) = 1 + x

Solution :

f o g (x)  =  f[g(x)]

We apply the function given for g(x).

f o g (x)  =  f[1 + x]

Instead of x, we have 1 + x, so we apply 1 + x instead of x in f(x).

f(1 + x)  =  4(1+x)2 - 1

  =  4(1 + x2 + 2x) - 1

  =  4 + 4x2 + 8x - 1

f o g (x)  =  4x2 + 8x + 3   ---(1)

g o f (x)  =  g[f(x)]

=  g [4x2 − 1]

Instead of x, we have 4x2 − 1, so we apply 4x2 − 1 instead of x in g(x).

 g [4x2 − 1]  =  1 + 4x2 − 1

  = 4x2  ---(2)

f o g (x)  g o f (x)

After having gone through the stuff given above, we hope that the students would have understood, "How to Find Composition of Two Functions". 

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