Question

Determine the range of the following expressions. (x² + x + 1)/(x² - x + 1)

Answer 1

Let y = (x² + x + 1)/(x² - x + 1)

⇒ x²y – xy + y = x² + x + 1

⇒ x²y – xy + y – x² – x – 1 = 0

⇒ x²(y – 1) – x(y + 1) + (y – 1) = 0

x is real ⇒ b² – 4ac ≥ 0

⇒ (y + 1)² – 4(y – 1)² ≥ 0

⇒ (y + 1)² – (2y – 2)² ≥ 0

⇒ (y + 1 + 2y – 2) (y + 1 – 2y + 2) ≥ 0

⇒ (3y – 1) (–y + 3) ≥ 0

⇒ –(3y – 1) (y – 3) ≥ 0

a = coeff. of y² = –3 < 0.,But

The expression ≥ 0

⇒ y lies between 1/3 and 3

∴ The range of (x² + x + 1)/(x² - x + 1) is [1/3, 3]