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.

Answer 1

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.