Question

Show
that f:R {-1}
R-{-1} given by f(x) = x/x+1 is invertible .Also find inverse of f​

Answer 1

Question :

Show that f : R - {-1} → R - { 1 } given by

f(x) = x/(x + 1) is invertible . Also find the inverse of f .

Answer :

Inverse function : g(x) = x/(1 - x)

Note :

★ A function f(x) is said to be invertible if it is one-one onto function .

★ One-One function : A function f(x) is said to be one-one if

f(x1) = f(x2) => x1 = x2 .

★ Onto function : A function f(x) is said to be onto function if Range = Co-domain .

Solution :

Given function :

f : R - {-1} → R - {1} , f(x) = x/(x + 1)

★ Whether f(x) is one-one :-

Let f(x1) = f(x2)

=> x1/(x1 + 1) = x2/(x2 + 1)

=> x1•(x2 + 1) = x2•(x1 + 1)

=> x1x2 + x1 = x2x1 + x2

=> x1 = x2

Since , f(x1) = f(x2) => x1 = x2 , thus the given function f(x) is one-one .

★ Whether f(x) is onto :-

Let y = f(x)

=> y = x/(x + 1)

=> y(x + 1) = x

=> yx + y = x

=> y = x - yx

=> y = x(1 - y)

=> x = y/(1 - y)

For x to be real , Denominator ≠ 0

=> 1 - y ≠ 0

=> -y ≠ -1

=> y ≠ 1

=> Range = R - { 1 }

=> Range = Co-domain

Since , Range = Co-domain thus the given function f(x) is onto .

→ f(x) is one-one onto function thus it is invertible .

→ The inverse of the given function f(x) will be ;

g(x) = x/(1 - x)