# Length Regular and Length Preserving

What does it mean to say a function is length regular and Length preserving? Does any one of them implies the other? Example if any could be useful

Length-regular: if $$x$$ and $$y$$ have the same length, then $$f(x)$$ and $$f(y)$$ have the same length.
Length-preserving: $$x$$ and $$f(x)$$ have the same length.
• $$f(x) = \overline{x}$$ (flip every bit in $$x$$): length-preserving.
• $$f(x) = x \|x$$ (concatenate $$x$$ with itself): length-regular but not length-preserving.