Question

Let \(f:R\rightarrow R\) be defined by \(f(x)=\frac{x^{2}+2x+3}{x^{2}+x+1}\). If the range of \(f(x)\) is \([a,b]\), then \(a+b=\ldots\)

• A. \(2\)
• B. \(4\)
• C. \(\frac{4}{\sqrt{3}}\)
• D. \(2\sqrt{3}\)

Hint:

Put y equal to the function, form a quadratic in x and require a non-negative discriminant.

Answer 1

The Correct Option is B

Step 1: Case y = 1
If $y=1$ the equation reduces to a linear one and gives a real x, so y = 1 lies in the range and is inside the interval.

Step 2: Use the roots
The inequality $3y^2-12y+8\le 0$ holds between its roots $\dfrac{6\pm 2\sqrt{3}}{3}$.

Step 3: Add them
$a+b=\dfrac{6-2\sqrt3}{3}+\dfrac{6+2\sqrt3}{3}=4$.

Final Answer:
The endpoints add up to 4. \[ \boxed{\text{(B) }4} \]