Implement monotonic x-y linear interpolation

Q: Implement monotonic x-y linear interpolation

This question evaluates a candidate's understanding of piecewise linear interpolation on strictly increasing x arrays, handling of out-of-range queries (error, clamp, or extrapolate), and the design of efficient interval-search strategies.

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Question

Implement a function interpolate(array_x, array_y, x) that returns the interpolated y-value at query point x.

You are given:

array_x : a strictly increasing (monotonic) list/array of floats (length n ≥ 2)
array_y : a list/array of floats of the same length as array_x
x : a float query that may be inside or outside the range [array_x[0], array_x[n-1]]

Requirements:

If x is within the domain, return the piecewise-linear interpolation between the two neighboring points.
Discuss and optionally support behaviors when x is out of range:
- Raise an error (no extrapolation)
- Clamp to the nearest endpoint
- Extrapolate linearly (e.g., extend the last segment)
Analyze time complexity and give at least one alternative to a binary-search (bisect) approach.

Assume array_x has no duplicates and both arrays fit in memory.

Implement monotonic x-y linear interpolation

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