Question

A function f : R `rightarrow` R defined by f(x) = 2 + x² is ______.

Options

• not one-one

• one-one

• not onto

• neither one-one nor onto

Answer 1

A function f : R `rightarrow` R defined by f(x) = 2 + x² is one-one.

Explanation:

Given, f(x) = 2 + x²

For one-one, f(x₁) = f(x₂)

`\implies 2 + x_1^2 = 2 + x_2^2` 

`\implies x_1^2 = x_2^2`

`\implies` x₁ = ±x₂

`\implies` x₁ = x₂

or x₁ = –x₂

Thus, f(x) is not one-one.

For onto

Let f(x) = y such that y ∈ R

= x² = y – 2

`\implies x = ±sqrt(y - 2)`

Put y = –3, we get

`x = ±sqrt(-3 - 2) = ±sqrt(-5)`

Which is not possible as root of negative is not a real number.

Hence, x is not real.

So, f(x) is not onto.