Let there be a function $f: \mathbb R \to \mathbb R$ given by $$f(x)=\begin{cases}5x + 2&x\geq 1\\x-1&x<1 \end{cases}$$ Give an example of a Left inverse of $f$, and prove that it is correct.
I can do this with normal function. But how does this work when we try a piece defined function?
Jasper Loy's answer here already shows you most of what you need in order to define $g:\Bbb R\to\Bbb R$ such that $g\circ f=id_{\Bbb R}$. It doesn't actually matter what $g(x)$ is for $0\le x<7,$ since $f$ never takes on values in that interval. Thus, we can define $g$ by $$g(x)=\begin{cases}\frac{x-2}5 & \text{for }x\ge 7\\x+1 & \text{for }x<0\\\text{just about any real-valued function you want} & \text{for }0\le x<7.\end{cases}$$
You can also take a look at this answer to see how one should proceed to construct a left inverse in general.