Boundaries for the domain of an absolute value function's inverse

I have the function $f(x) = \mid 2x-1 \mid$, and I'm trying to find its inverse function(s).

I found an inverse relationship $\frac {1 \pm x}{2}$. In order to get inverse functions out of this relationship, I believe the next step would be to determine a restricted domain on which the relationship would give a function.

Is it true that $f(x)$ has inverses $f^{-1}(x) = \frac {1 + x}{2}$ for $x \geq -1$ and $f^{-1}(x) = \frac {1 - x}{2}$ for $x \leq 1$. If so, how could I check?

I think this works because each of these inverses is a function (i.e. the plot of each inverse, with its respective boundary, passes the "horizontal line test" for inverse functions).

Thanks for any guidance you could provide!

• Hint: where is the vertex of the function's graph? – MathematicsStudent1122 Dec 2 '16 at 4:19
• Thank you, @MathematicsStudent1122. Since the vertex of this function is 1, is it true that the vertex of any absolute value function indicates how to put boundaries on the function's inverse? – sawghol Dec 2 '16 at 4:22