FIND COMPOSITION OF TWO FUNCTIONS

Definition : 

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.

Note : 

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

Example 1 :

Find f o g, if f(x) = x and g(x) = 3x2.

Solution :

f o g = f[g(x)]

f[3x2]

= 3x2

Example 2 :

If f(x) = 2x - 5 and g(x) = x + 2 find f o g.

Solution :

f o g = f[g(x)]

f[x + 2]

= 2(x + 2) - 5

= 2x + 4 - 5

= 2x - 1

Example 3 :

If f(x) = -3x + 4 and g(x) = x2 find g o f. 

Solution :

g o f = g[f(x)]

= g[-3x + 4]

= (-3x + 4)2

= (-3x)2 + 2(-3x)(4) + 42

= 9x2 - 24x + 16

Example 4 :

If f(x) = 4x2 + 2 and g(x) = √(x- 8) find f o g.

Solution :

f o g = f[g(x)]

= f[√(x- 8)]

= 4[√(x- 8)]2 + 2

4(x- 8) + 2

= 4x - 32 + 2

= 4x - 30

Example 5 :

If f(x) = -9x + 4 and g(x) = x4, find g o f.

Solution :

g o f = g[f(x)]

= g[-9x + 4]

= (-9x + 4)4

Example 6 :

If f(x) = 2x + 1 and g(x) = x2 - 2, then verify f o g = g o f.

Solution :

f o g :

= f[g(x)]

f[x2 - 2]

2(x2 - 2) + 1

= 2x2 - 4 + 1

= 2x2 - 3 ----(1)

g o f :

= g[f(x)]

= g[2x + 1]

(2x + 1)2 - 2

= (2x)2 + 2(2x)(1) + 12 - 2

= 4x2 4x + 1 - 2

= 4x2 4x - 1 ----(2)

From (1) and (2), we see that f o g ≠ g o f.

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