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Two functions f and g are called identical (equal) only if they have the same domain, the same co-domain, and f(x) = g(x) for every x in the common domain (same graph). For what interval of x are f(x) = log(x - 1) - log(x - 2) and g(x) = log((x - 1)/(x - 2)) identical?

1. [1, 2]
2. [2, infinity)
3. (2, infinity)
4. (-infinity, infinity)

Correct answer: (2, infinity)

Solution

f is defined only where both logs exist (x > 2). g is defined wherever the ratio is positive (x < 1 or x > 2). The functions agree only on the intersection of their domains, which is x > 2.

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