Question

The range of the function \[ f(x)=\frac{x^2+x+1}{x^2-x+1} \] is

• A. \(\left[\frac{1}{3},3\right]\)
• B. \(\left[\frac{1}{2},2\right]\)
• C. \(\left[-\frac{1}{2},-\frac{1}{4}\right]\)
• D. \(\left[-\frac{1}{2},2\right]\)

Hint:

To find the range of a rational function, put \(y=f(x)\), form a quadratic equation in \(x\), and use the condition discriminant \(\geq 0\).

Answer 1

The Correct Option is A

Step 1: Check boundary values and monotonicity.
At \(x=0\): \(f(0)=1\). As \(x\to\infty\): \(f(x)\to 1\). At \(x=1\): \(f(1)=3/1=3\). At \(x=-1\): \(f(-1)=1/3\).

Step 2: Confirm the range by finding critical points.
Set \(y=\frac{x^2+x+1}{x^2-x+1}\), cross-multiply: \((y-1)x^2-(y+1)x+(y-1)=0\). For real \(x\), discriminant \(\geq0\): \((y+1)^2-4(y-1)^2\geq0\Rightarrow(y+1-2y+2)(y+1+2y-2)\geq0\Rightarrow(3-y)(3y-1)\geq0\Rightarrow y\in[1/3,3]\). \[ \boxed{\left[\dfrac{1}{3},3\right]} \]