It seems like you are looking for a specific x value that satisfies the equation f(x) = y, where y is a constant and f(x) is a non-linear function with no other variables involved. Is my interpretation correct?
Assuming that f(x) is continually increasing or decreasing without any fluctuations, a binary search method can be used to pinpoint the exact solution:
Determine values
x_lo such that f(x_lo) < yx_hi such that f(x_hi) > y
Calculate the midpoint m = (x_hi + x_lo) / 2.
- If
f(m) < y, update x_lo = m, and repeat this step
- If
f(m) > y, update x_hi = m, and repeat this step
- If
f(m) == y, or it is sufficiently close to your requirement, you have found the desired result!
An initial approach to determining values for x_hi and x_lo is by testing the f(any) value and assigning x_hi=any or x_lo=any based on its relationship with y. Once one extreme is identified, the opposite extreme can be easily found by progressively increasing or decreasing it. For example, if we have x_lo, we can evaluate using x_hi = x_lo*(2^i) for i=1, 2, 3, ..., where ^ denotes exponentiation.
This method may not be effective when f(x) intersects with y at multiple points since it might skip over the crucial x in the second step of the process.