Question

What will be the range of the expression \(x - 2\over x^2 + x + 3\), where x is real ?
1. (-1, 1/11]
2. [1, 1/11]
3. (1, -1/11]
4. [-1, 1/11]

Answer 1

Correct Answer - Option 4 : [-1, 1/11]

Calculation :

Let f(x) = \(x - 2 \over x^2 + x + 3\)  = y

⇒ yx² + yx + 3y - x + 2 = 0

⇒ yx² + (y - 1)x + 3y + 2 = 0

f(x) can have any value y, provided the above equation in x has real roots.

∴ b² - 4ac ≥ 0

⇒ (y - 1)² - 4y(3y + 2) ≥ 0

⇒ -11y² - 10y + 1 ≥ 0

⇒ 11y2 + 10y - 1 ≤ 0

⇒ (11y - 1)(y + 1) ≤ 0 

⇒ -1 ≤ y ≤ \(1\over 11\)

So, the range of y or f(x) is [-1, 1/11]

 The Domain of a function f(x) is the set of all the values for which the function is defined, and the Range is the set of all the values that function f(x) takes.