Question

Answer the following:

Find the range of the following function.

f(x) = `x/(9 + x^2)`

Answer 1

f(x) = `x/(9 + x^2)`

Let y = f(x)

∴ y = `x/(9 + x^2)`

∴ yx² + 9y = x

∴ yx² – x + 9y = 0

This is a quadratic equation in x

∵ x is real

∴ Δ ≥ 0

∴ ( – 1)² – 4(y)(9y) ≥ 0

∴ 1 – 36y² ≥ 0

∴ 36y² ≤ 1

∴ `y^2 ≤ 1/36`

∴ `-1/6 ≤ y ≤ 1/6`

∴ Range = `[-1/6, 1/6]`.