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  • Let the function f: R → R be defined by f (x) = cos x, ∀ x ∈ R. Show that f is neither one-one nor onto.

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

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It is given that, f: R \rightarrow R, f(x) = cos x, \forall x \in R\\

So, we can write:

f(x\textsubscript{1}) = f(x\textsubscript{2})\\ or, cos x\textsubscript{1}= cos x\textsubscript{2}\\

Hence, x\textsubscript{1 }= 2n \pi \pm x\textsubscript{2}, where n \in Z\\

It is understandable that for any value of x\textsubscript{1 }and\: \: x\textsubscript{2}, the above equation has an infinite number of solutions.

Therefore, f(x) is a many one function.

We know the range of cos x is [-1, 1] and it is a subset of the given co-domain R.

Hence, the given function is not onto.

Posted by

infoexpert22

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