Without additional constraints, this is not true. We will define a different function that satisfies [math]f(f(x)) = x + 1[/math].
Let us start by defining an auxiliary function [math]g(x)[/math] for [math]x \in [-\frac12,\frac12][/math]
[math]g(x) = \displaystyle\frac{12x+\left|10x-3\right| - \left|10x+3\right|}{24} \tag*{}[/math]
The graph of [math]g(x)[/math] looks like this
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