Your book probably says: if \( X \) has a standard (continuous) uniform distribution (on \( (0, 1) \)), then Y = 1 - X has the same distribution.

There are standard techniques to find the distribution of transformation of random variables like this. Look for the CDF method or Jacobian method.

You should not transform the pdf like this f_{1-X}(x) \neq 1 - f_X(x) . It is not the way to obtain the pdf of the transformed random variable. And the pdf you obtained make no sense as the area under it is \( 0 \)