**Evaluate the Missing Value Using Composition of Two Functions :**

Here we are going to see, how to evaluate the missing value using 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.

**Question 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]

Now we apply (x + 1)/2 instead of x in f(x).

= 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)

(1) = (2)

**Question 2 :**

(i) If f (x) = x^{2} −1, g(x) = x −2 find a, if g o f (a) = 1 .

**Solution :**

f(a) = a^{2} −1 and g(a) = a −2

g o f (a) = g[f(a)]

= g[a^{2} −1]

= (a^{2} −1) - 2

g o f (a) = a^{2} −3

Given that

g o f (a) = 1

a^{2} −3 = 1

a^{2} = 4

a = ±2

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

**Solution :**

**f o f (k) = f[f(k)]**

** = f[2k - 1]**

Now we apply 2k - 1 instead of k in f(k)

= 2(2k - 1) - 1

= 4k - 2 - 1

**f o f (k)** = 4k - 3

Given that :

**f o f (k) = 5**

**4k - 3 = 5**

**4k = 5 + 3**

**4k = 8**

k = 2

Hence the value of k is 2.

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