HOW TO FIND THE INVERSE OF A FUNCTION USING COMPOSITION
About "How to find the inverse of a function using composition"
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).
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