Evaluate piecewise linear function at x

You are given a polyline defined by n 2D points $(x_i, y_i)$. Connecting consecutive points with straight line segments forms a piecewise linear function.

Task

Given a target value x, return the corresponding function value y(x):

• If x equals some x_i , return y_i .
• If x lies strictly between x_i and x_{i+1} , linearly interpolate on the segment between $(x_i, y_i)$ and $(x_{i+1}, y_{i+1})$ :

$y(x)=y_i + (y_{i+1}-y_i)\cdot\frac{x-x_i}{x_{i+1}-x_i}$

• If x < x_0 or x > x_{n-1} , return null (or a sentinel) because the function is undefined outside the polyline.

Input

• points : list of n pairs (x, y)
• x : target x-coordinate

Assumptions / Constraints

• n >= 2
• Points are given sorted by strictly increasing x (i.e., x0 < x1 < ... < x(n-1) ).

Output

• A numeric value y(x) (float), or null if out of range.