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

Dear Student

We have,
f: R → R, f(x) = cos x
Now,
f (x1) = f (x2)
cos x1= cos x2
x1=2nπ±x2,n∈Z
It’s seen that the above equation has infinite solutions for x1and x2Hence, f(x) is many one function.
Also the range of cos x is [-1, 1], which is subset of given co-domain R.
Thus,the given function is not onto.
Thanks