# For the function f(x) = (2 - x)/(2x - 1) what is the inverse function f-1(x). Also, what is the domain and range of both the functions.

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### 1 Answer

The function `f(x) = (2- x)/(2x - 1)` . The function f(x) and its inverse `f^-1(x)` are related by `f(f^-1(x)) = x` .

`f(f^-1(x)) = x` .

=> `(2 - f^-1(x))/(2*f^-1(x) - 1) = x`

=> `2 - f^-1(x) = x*(2*f^-1(x) - 1)`

=> `2 - f^-1(x) = 2*x*f^-1(x) - x`

Bring `f^-1(x)` to one side and take x to the other side.

=> `f^-1(x)(2x + 1) = 2 + x`

=> `f^-1(x) = (2 + x)/(2x + 1)`

The domain of a function is all the values of x for which the function is defined.

For `f(x) = (2 - x)/(2x - 1)` , f(x) is defined for all values except x = 1/2 at which point the denominator is 0. The domain of this function is R - {1/2}. The range of the function is R - {-1/2}

For the inverse function f^-1(x), the domain is R - {-1/2} and the range is R - {1/2}

`f(x) = (2- x)/(2x - 1)`

`f^-1(x) = (2 + x)/(2x + 1)`