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Let f : R rarr R be defined by f(x) = co...

Let `f : R rarr R` be defined by `f(x) = cos (5x+2)`. Is f invertible? Justify your answer.

Text Solution

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For invertible of f, f must be bijective (i.e, one-one onto).
If `x_(1), x_(2) in R`,
then `f(x_(1))=f(x_(2))`
`implies cos(5x_(1)+2)=cos(5x_(2)+2)`
`implies 5x_(1)+2=2npipm(5x_(2)+2)`
`implies x_(1)nex_(2)`
`therefore` f is not one-one.
But `-1lecos(5x+2)le1`
`therefore -1lef(x)le1`
Range `=[-1,1]subR`
`therefore` f is into mapping.
Hence, the function f(x) is no bijective and so it is not invertible.
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