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One - One function

Which of the ...

Question

Which of the following pairs are equal functions?
(1) $ln x^{2}, 2 ln x; x > 0$
(2) $\frac{x^{2}}{x}, x$
(3) $∣ x ∣^{2}, x^{2}$
(4) $∣ x ∣^{3}, x^{3}$

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Solution

1) For $x > 0$ , $2 ln x = ln x^{2}$ .
2) $\frac{x^{2}}{x} = x$ only for $x \neq 0$ .
3) $| x |^{2} = x^{2}$ , $\forall x \in R$ .
4) $| x |^{3} = x^{2} | x | = x^{3}$ only for $x > 0$ .
So, there are two pairs which are same and they are (1) and (3).

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