How to find the inverse of the Piecewise Linear function

Question

I have the following function:

$k_1x + b, x < x_0$

$k_1x + b + k_2(x-x_0), x_0 \leq x < x_1$

$k_1x + b + k_2(x-x_0) + k_3(x - x_1), x \geq x_1$

So the parameters are:

$x_0, x_1, b, k_1, k_2, k_3$

I would like to invert this function changing the values of the parameters.

One idea is to invert the slopes $k_1$, $k_2$, $k_3$.

Would that be sufficient?

Thanks.

Answer 1

Hint:

In the three intervals, you have a relation of the form

$$y=ax+b,$$

which is inverted as

$$x=\frac{y-b}a.$$

What you have to do is

• to find the $y_k$ values corresponding to the $x_k$, which delimit the $y$ intervals,

• to obtain the $a_k,b_k$ coefficients in the three pieces.