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I want to show the function $ \tan(\pi x - \frac \pi2) $ is one-to-one if x $ \in (0,1) $. But an argument I would normally use to prove a function is one-to-one (letting $ \tan(\pi x_1 - \frac \pi2) $ = $ \tan(\pi x_2 - \frac \pi2) $ and then showing $ x_1 $ = $ x_2 $) doesn't seem to work.

I was hoping someone could give me any suggestions.

Thanks.

2 Answers 2

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The derivative is defined (exists) and positive on $(0.1)$, so the function is increasing over the interval.

Remark: The argument does not work over larger intervals, since then the derivative does not always exist, although it is always positive where it exists.

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If $x\in(0,1)$, then $0