PROBLEMS ON COMPOSITION OF FUNCTIONS

Problem 1 :

If f(x) = 2x - 1, g(x) = (x + 1)/2, show that

f o g = g o f = x

Solution :

f o g = f[g(x)]

= f[(x + 1)/2]

= 2((x + 1)/2) - 1

= x + 1 - 1

= x ----(1)

g o f = g[f(x)]

=  g[2x - 1]

= (2x - 1 + 1)/2

= 2x/2

  = x ----(2)

From (1) and (2), 

f o g = g o f = x

Problem 2 :

If f(x) = x2 - 1, g(x) = x - 2 find a, if g o f(a) = 1.

Solution :

g o f(a) = 1

g[f(a)] = 1

g[a2 - 1] = 1

a2 - 1 - 2 = 1

a2 - 3 = 1

a2 = 4

Take square root on both sides. 

a = ±2

Problem 3 : 

Find k, if f(k) = 2k - 1 and f o f(k) = 5.

Solution :

f o f(k) = 5

f[f(k)] = 5

f[2k - 1] = 5

2(2k - 1) - 1 = 5

4k - 2 - 1 = 5

4k - 3 = 5

4k = 8

k = 2

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