Let the function f: R → R be defined

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

Solution:

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 x1 and x2

Hence, f(x) is many one function.

Also the range of cos x is [-1, 1], which is subset of given co-domain R.

Therefore, the given function is not onto.