Question

Let f: R→Rbe defined as f (x) = `(x^2 + 1)/2`, then ______.

Options

• f is one-one onto

• fis one-one but not onto

• fis onto but not one-one

• fis neither one-one nor onto

Answer 1

Let f: R→Rbe defined as f (x) = `(x^2 + 1)/2`, then f is neither one-one nor onto.

Explanation:

The function f (x) = `(x^2 + 1)/2` is not one-to-one because it is a symmetric parabola where different inputs can yield the same output; for instance, both x = 1 and x = −1 result in f(x) = 1. It is also not onto because the squared term x² can never be negative, which restricts the function's range to values ≥ 0.5. Since the function never reaches negative numbers or any value below 0.5, it fails to cover the entire codomain of all real numbers.