I'm having trouble envisioning a bijective relationship that maps $(0,1)$ to $(0,1]$.

My professor gave the hint that it can be expressed as a piece-wise function $f(x)$ comprising of two cases: **\_ if $x=$ \_\_** and $x$ otherwise.

It could be $1$ if $x=\frac{1}{2}$ and $x$ if $0<x<\frac{1}{2}$ or $\frac{1}{2}<x<1$.

However, $f$ would not be a bijection then, because there would not be an $x$ value for $y=\frac{1}{2}$.