**How to find the inverse of a function using composition :**

f(x) and g(x) are inverse functions on the set of x-values where their compositions are defined if and only if the following equations are true:

(f∘g) (x) = f (g(x)) = x --(1)

(g∘f) (x) = g (f(x)) = x ---(2)

Let us see some example problems to understand the above concept.

**Example 1 :**

Is g(x) the inverse function of f(x) ?

f(x) = x – 3 and g(x) = 4x + 8

**Solution :**

To check whether g(x) is the inverse of f(x), we have to find f∘g (x) and g∘f (x)

f∘g (x) = f [g(x)]

= f [4x + 8]

instead of x in the function f(x), we have to apply 4x + 8.

= 4 x + 8 - 3

= 4 x + 5 ≠ x

Since f∘g (x) is not equal to x, we don't have to find the value of g∘f (x).

Hence g(x) is not inverse of f(x).

**Example 2 :**

Is g(x) the inverse function of f(x) ?

f(x) = x – 3 and g(x) = -5x

**Solution :**

To check whether g(x) is the inverse of f(x), we have to find f∘g (x) and g∘f (x). First let us find the value of f∘g (x).

f∘g (x) = f [g(x)]

= f [-5x]

instead of x in the function f(x), we have to apply -5x.

= -5x - 3 ≠ x

Since f∘g (x) is not equal to x, we don't have to find the value of g∘f (x).

Hence g(x) is not inverse of f(x).

**Example 2 :**

Is f(x) the inverse function of g(x) ?

f(x) = x and g(x) = 3 x + 8

**Solution :**

To check whether g(x) is the inverse of f(x), we have to find f∘g (x) and g∘f (x). First let us find the value of f∘g (x).

f∘g (x) :

f∘g (x) = f [g(x)]

= f [3x + 8]

instead of x in the function f(x), we have to apply 3x + 8.

= 3x + 8 ≠ x ---(1)

Since f∘g (x) is equal to x, we have to find the value of g∘f (x).

g∘f (x) :

g∘f (x) = g [f(x)]

= g [x]

instead of x in the function g(x), we have to apply x.

= 3x + 8 ≠ x ---(2)

Hence g(x) is not the inverse of f(x).

After having gone through the stuff given above, we hope that the students would have understood "How to find the inverse of a function using composition".

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