f (x) = cosx, ∀ x ∈ R
In order to prove that f is one-one, it is sufficient to prove that f(x1)=f(x2) ⇒ x1=x2∀ x1, x2 ∈ A .
Let x1 = 0 and x2 = 2π are two different elements in R.
f(x1) = f(0) = cos0 = 1
f(x2) = f(2π) = cos2π = 1
we observe that f(x1)=f(x2) but x1 ≠ x2.
This shows that different element in R may have same image.
Thus, f(x) is not one-one.
We know that cosx lies between -1 and 1.
So, the range of f is [-1,1] which is not equal to its co-domain.
i.e., range of f ≠ R (co-domain)
In other words, range of f is less than co-domain, i.e there are elements in co-domain which does not have any pre-image in domain.
so, f is not onto.
Hence, f is neither one-one nor onto.
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