Need help with mathematical notation in defining a line

Question

What I'm trying to do is describe a generalized form of the following piecewise function using proper notation.

What I have so far is (excuse my non-existent latex skills):

• f(x) is defined between [0, 1]

• f(x; A, B, C) -> the function is parameterized by A, B, C (three arbitrarily named variables representing, respectively, the y-value of the flat portion of the function, the x-value at which the function takes on that sloping function, and C is the y-value at f(1.0)

f(x) = {
   A; if X < B
   [the linear function]; if X >= B
}

That is, the two components are f(x < B) = A and f(x >= B) = the line containing the points f(B) = 1, f(1) = C. The graph above is not correct because there is a discontinuity at f(B), so ignore that.

Thanks!

Answer 1

This function is

$$f(x)=\begin{cases} a,&0\leq x<b\\ \frac{c-1}{1-b}(x-b)+1,&b\leq x\leq 1 \end{cases}$$

Given two points $(x_1,y_1)$ and $(x_2,y_2)$, the line passing through them is $y-y_1 = \frac{y_2-y_1}{x_2-x_1}(x-x_1)$; this is where the linear function for $x \in [b,1]$ comes from.