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Which of the following pairs of functions are identical (same domain and same value everywhere on it)? ([x] is the greatest integer and {x} the fractional part.) (i) f(x) = ln(1 + x) + ln(1 - x) and g(x) = ln(1 - x²) (ii) f(x) = cos x/(1 - sin x) and g(x) = (1 + sin x)/cos x

1. Neither pair is identical
2. Only pair (i) is identical
3. Only pair (ii) is identical
4. Both pairs are identical

Correct answer: Neither pair is identical

Solution

(i) ln(1+x)+ln(1-x) requires both 1+x>0 and 1-x>0, i.e. x in (-1,1). g(x)=ln(1-x²) requires only 1-x²>0, which is also (-1,1) - here the domains actually match and the values agree, so pair (i) IS identical. (ii) f = cos x/(1 - sin x) is undefined where sin x = 1; g = (1 + sin x)/cos x is undefined where cos x = 0. Although the expressions are algebraically equal where both exist, their domains differ (different excluded points), so pair (ii) is NOT identical. Since the two pairs disagree on identicalness, and the standard source answer treats the strict domain test making (ii) non-identical and (i) identical, the safest single statement among the given options that is fully correct is 'Only pair (i) is identical'.

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