The function fx x2 34x2 x 0 is onetoone Find its inverse

The function f(x) = x^2 - 3/4x^2, x > 0, is one-to-one. Find its inverse and check your answer.

Solution

Given that

f(x) = (x² - 3 ) / 4x² , x >0 ,is one to one

We know that

Inverse function :    

Let f be a one-to-one function with domain A and range B.

Then its inverse function, denoted f ¹ , has domain B and range A and is defined by f ¹ (y) = x if and only if f(x) = y for any y in B.

  f(x) = (x² - 3 ) / 4x²

y = (x² - 3 ) / 4x²

ineterchange x and y

x =  (y² - 3 ) / 4y²

x = ( y ² / 4y² ) - ( 3/4y² )

x = 1/4 - 3/4y²

x - 1/4 = -3/4y²

   ( x - 1/4 ) /(-4/3) = y²

( 4x - 1 ) / 3 = y²

[ ( 4x - 1 ) / 3 ]^1/2 = y

Hence,

f^-1(x) =   [ ( 4x - 1 ) / 3 ]^1/2