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aev [14]
3 years ago
11

PLEASE HELP ASAP NOW WILL MARK BRAINIEST LOTS OF POINTS!!!!

Mathematics
1 answer:
Ira Lisetskai [31]3 years ago
4 0

22. On its own, \sin x is not invertible because it's not one-to-one because it's periodic. For instance, we can always find more than one value of x for which \sin x=0; this happens when x=0,\pm\pi,\pm2\pi,\ldots.

But we can restrict the domain so that it can become invertible. If we only allow values of x within -\dfrac\pi2\le x\le\dfrac\pi2, for example, then each x will only be associated with a single value of \sin x. This is how the standard inverse sine is defined. With -\dfrac\pi2\le x\le\dfrac\pi2 (restricted domain of sine), we guarantee that -1\le\sin x\le1 (range). So the domain of the inverse is -1\le x\le 1, and the range of the inverse is -\dfrac\pi2\le\sin^{-1}x\le\dfrac\pi2.

\dfrac{3\pi}4 is not in this restricted domain. But \sin\dfrac{3\pi}4 still exists as long as we take the standard domain (the entire real line), and \sin\dfrac{3\pi}4=\dfrac1{\sqrt2}. But then \sin^{-1}\left(\sin\dfrac{3\pi}4\right)=\dfrac\pi4 because this is the only value of x for which \sin^{-1}x=\dfrac1{\sqrt2}.

In short: \sin and \sin^{-1} are NOT inverses of one another, but rather one is an imperfect inverse of the other.

25. Not much to say about this:

g(x)=\cos x\implies g\left(\dfrac\pi2\right)=\cos\dfrac\pi2=0

f(x)=\sin x\implies f\left(g\left(\dfrac\pi2\right)\right)=\sin\left(\cos\dfrac\pi2\right)=\sin0=0

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