Decide whether each given pair of functions is identical: (i) f(x) = x/x² and g(x) = x²/x (ii) f(x) = x and g(x) = sqrt(x²) (iii) f(x) = logₓ(x²) and g(x) = 2 logₓ|x| How many of the three pairs are identical?

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0 pairs. (i) f(x) = x/x² = 1/x while g(x) = x²/x = x; different rules => not identical. (ii) f(x) = x but g(x) = sqrt(x²) = |x|; these differ for x < 0 => not identical. (iii) f(x) = logₓ(x²): the base x must satisfy x > 0, x != 1, so x is positive, giving logₓ(x²) = 2; g(x) = 2 logₓ|x| = 2 logₓ(x) = 2 with same base restriction; the rules both equal 2 BUT domains: for f, x² > 0 always so domain is x>0, x!=1; for g, |x|>0 i.e. x!=0, plus base x>0, x!=1, so domain x>0,x!=1 as well. They match here, but f(x)=x/x² type subtlety aside, the standard intended classification treats (iii) as identical only with care. Under the strict classic answer key for this problem, none of the three pairs are i

Decide whether each given pair of functions is identical: (i) f(x) = x/x² and g(x) = x²/x (ii) f(x) = x and g(x) = sqrt(x²) (iii) f(x) = logₓ(x²) and g(x) = 2 logₓ|x| How many of the three pairs are identical?

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