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Find the domain and range of the functio...

Find the domain and range of the function `f(x)=1/(2-sin3x)` .

Text Solution

Verified by Experts

The correct Answer is:
dom `(f)=R," range "(f)=[(1)/(3),1]`
Clearly, `-1le-sin3xle1" for all "x""inR`
`implies2+(-1)le2+(-sin3)le2+1" for all "x""inR`
`implies1le2sin3xle3" for all "x""inR`
`implies2-sin3xne0" for any "x""inR`.
`:.f(x)=(1)/(2-sin3x)` is defined for all `x""inR`.
We have, `2-sin3xge1and2-sin3xle3`
`:.(1)/(2-sin3x)le1and(1)/(2-sin3x)ge(1)/(3)implies(1)/(3)le(1)/(2-sin3x)`.
`:."range "(f)=[(1)/(3),1]`.
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