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Let the function f:R to R be defined by ...

Let the function `f:R to R` be defined by `f(x)=cos x, AA x in R.` Show that `f` is neither one-one nor onto.

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Let `x_(1), x_(2) in R`.
Then, `f(x_(1))=f(x_(2))impliescosx_(1)=cosx_(2)` implies `x_(1)=2npipmx_(2)impliesx_(1)nex_(2)`
`therefore` f is not one-one.
Let `y=cosx,"but"-1lecosxle1`
`therefore yin[-1,1]`
`[-1,1]subR`
So, f is into (not onto).
Hence, f is neither one-one nor onto.
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