Question

Let f: R `rightarrow` R be defined by f(x) = `x/(1 + x^2), x ∈ R`. Then the range of f is ______.

Options

• `[-1/2, 1/2]`

• R – [–1, 1]

• `R - [-1/2, 1/2]`

• (–1, 1) – {0}

Answer 1

Let f: R `rightarrow` R be defined by f(x) = `x/(1 + x^2), x ∈ R`. Then the range of f is `underlinebb([-1/2, 1/2]`.

Explanation:

f(x) = `x/(1 + x^2), x ∈ R`

Let y = f(x)

∴ y = `x/(1 + x^2)`

⇒ yx² − x + y = 0

⇒ x = `(1 ± sqrt(1 - 4y^2))/(2y)`

For x to be defined,

1 − 4y² ≥ 0, y ≠ 0

⇒ `y ∈ [-1/2, 1/2] - {0}`

But for x = 0, y = 0

∴ R(f) = `[-1/2, 1/2]`