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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?

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3 Answers

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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: $$ \begin{align} &y= \begin{cases} 12x+3, & \text{if $x\ge0$} \\ x+3, & \text{if $x\lt0$} \end{cases}\\ &\implies \begin{cases} y=12x+3, & \text{if $x\ge0$} \\ y = x+3, & \text{if $x\lt0$} \end{cases}\\ &\implies \begin{cases} y=12x+3, & \text{if $12x+3\ge 3$} \\ y = x+3, & \text{if $x+3\lt 3$} \end{cases}\\ &\implies \begin{cases} y=12x+3, & \text{if $y\ge 3$} \\ y = x+3, & \text{if $y\lt 3$} \end{cases}\\ &\implies \begin{cases} x=\frac{y-3}{12}, & \text{if $y\ge 3$} \\ x=y-3, & \text{if $y\lt 3$} \end{cases}\\ &\implies x=\begin{cases} \frac{y-3}{12}, & \text{if $y\ge 3$} \\ y-3, & \text{if $y\lt 3$} \end{cases}\\ \end{align} $$

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The first one is correct.

For $x<0$ it is $x-3$.

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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 $x >3$

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