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Algebra Difficulty 4.9 AIME Find the answer

Example 14(Ai). Are the functions (1) y=eln⁡(x2−2x−3)y=e^{\ln \left(x^{2}-2 x-3\right)} and (2) y=(x2−2x−3)2y=\sqrt{\left(x^{2}-2 x-3\right)^{2}} the same?

A number or a short expression. Spacing and $ signs are ignored.

Solution

Simplify: (1) y=x2−2x−3y=x^{2}-2 x-3, (2) y=x2−2x−3y=x^{2}-2 x-3, the domain for both is x∈Rx \in R, therefore they are the same function.

Analysis: (1) From x2−2x−3>0\mathrm{x}^{2}-2 \mathrm{x}-3>0 find the domain of function (1) (−∞,−1)∪(3,+∞0)3(-\infty,-1) \cup(3,+\infty 0)_{3}, the domain is R, hence they are different. Additionally, the ranges of (1), (2), and y=x2−2x−3y=x^{2}-2 x-3 are also different. Therefore, when transforming, pay attention to the conditions implicitly contained in concepts, formulas, and rules to prevent errors.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.