Problem: Express the implicit curve given by (y − 1)² = 4 − ln(x) as a function y = y(x) on some neighborhood of the point (1,−1).

Solution: Is this a good question or a trick question? We start by checking that the given point really satisfies the given equality: Check. So at least this makes sense, the point lies on the given curve.

We could try to use theory (Implicit function theorem, partial derivatives) to prove that finding a function is possible, but it would not help us actually solving the problem. They want us to find a concrete function, so unless this is a trick question, such a thing is possible. Therefore we simply try it and we will see whether it works. We will try to express y from the given equation.

We start by taking the square root on both sides of the equality and won't forget to put both possible signs.

Which sign is the right one? We know that the function we obtain must work for the point (1,−1). Thus we try to substitute to both versions above (with plus and with minus) the values x = 1 and y = −1. We see that the "plus" variant does not work, while the "minus" one does. This decides which way we should go.

On what neighborhood of x = 1 does this work? Logarithm requires x > 0, the square root forces us to take x ≤ e⁴. Thus we have the neighborhood U = (0,e⁴).

We check that for x from U, if we substitute this y(x) for y into the given equation above, it will be true.

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