Inverse of piecewise function I've the following function:
$$f(x)= 
\begin{cases}
12x+3, & \text{if $x\ge0$} \\
x+3, & \text{if $x\lt0$}
\end{cases}
$$
What will be its inverse?
For me is $f(x)^{-1}= \frac{x-3}{12}$ per $x\ge 0$ and $3-x$ for $x\lt 0$. Right? 
 A: To find the inverse of a function, we set $y=f(x)$ and try to solve for $x$ in terms of $y$. You can do this by direct manipulation of each case:
$$
\begin{align}
&y= 
\begin{cases}
12x+3, & \text{if $x\ge0$} \\
x+3, & \text{if $x\lt0$}
\end{cases}\\
&\implies
\begin{cases}
y=12x+3, & \text{if $x\ge0$} \\
y = x+3, & \text{if $x\lt0$}
\end{cases}\\
&\implies
\begin{cases}
y=12x+3, & \text{if $12x+3\ge 3$} \\
y = x+3, & \text{if $x+3\lt 3$}
\end{cases}\\
&\implies
\begin{cases}
y=12x+3, & \text{if $y\ge 3$} \\
y = x+3, & \text{if $y\lt 3$}
\end{cases}\\
&\implies
\begin{cases}
x=\frac{y-3}{12}, & \text{if $y\ge 3$} \\
x=y-3, & \text{if $y\lt 3$}
\end{cases}\\
&\implies
x=\begin{cases}
\frac{y-3}{12}, & \text{if $y\ge 3$} \\
y-3, & \text{if $y\lt 3$}
\end{cases}\\
\end{align}
$$
A: The first one is correct.
For $x<0$ it is $x-3$.
A: Try to draw the function as a graph, in a sheet of paper, and rotate it so to invert axes. It becomes obvious that the breaking point is not $x = 0$ anymore but becomes $x = 3$. This is because in the original function at $x=0$ corresponds $y = 3$, and in the inverse the opposite is true. The correct result is thus
$x - 3$ for $x < 3$ and $\frac{(x - 3)}{12}$ for $x >3$
