Question

Range of the function f(x) = `x/(1 + x^2)` is ______.

Options

• (– ∞, ∞)

• [– 1, 1]

• `[-1/2, 1/2]`

• `[-sqrt(2), sqrt(2)]`

Answer 1

Range of the function f(x) = `x/(1 + x^2)` is `underlinebb([-1/2, 1/2]`.

Explanation:

Let y = `x/(1 + x^2)`

`\implies` x²y – x + y = 0

For x to be real, 1 – 4y² > 0  {Discriminant = 1 – 4y²}

`\implies` 4y² – 1 ≤ 0

`\implies` (2y – 1)(2y + 1) ≤ 0

`\implies - 1/2 ≤ y ≤ 1/2`

∴ y = `f(x) ∈ [-1/2, 1/2]`