Question

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Determine the range of
(x² + x + 1)/(x² - x + 1)​

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Answer 1

Answer:

Let y = (x²+x+1)/(x²−x+1)

⇒yx²-yx+y=x²+x+1

⇒(x−1)x²+(y+1)x+(y−1)=0

If x∈R, then 

Discriminant ≥0

⇒(y+1)²−4(y−1)² ≥ 0

⇒−3y²+10y−3 ≥ 0

⇒3y²−10y+3 ≤ 0

⇒(3y−1)(y−3) ≤ 0

⇒31 ≤ y ≤ 3

∴ Range =[1/3,3]

Explanation:

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Answer 2

So, The Range of ( x²+ x + 1 ) / ( x² - x + 1 ) is [ 1/3 , 3 ]