Question

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

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Answer 1

Answer:

Let y=

x

2

+x+1

x

2

−x+1

⇒yx

2

+yx+y=x

2

−x+1

⇒(x−1)x

2

+(y+1)x+(y−1)=0

If x∈R, then

Discriminant ≥0

⇒(y+1)

2

−4(y−1)

2

≥0

⇒−3y

2

+10y−3≥0

⇒3y

2

−10y+3≤0

⇒(3y−1)(y−3)≤0

⇒

3

1

≤y≤3

∴ Range =[

3

1

,3]

Answer 2

Explanation:

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To determine the range of the function f(x) = (x² + x + 1)/(x² - x + 1), we need to find the values that f(x) can take.

To start, note that the denominator of f(x) is always positive because it represents a quadratic expression. Let's factorize the denominator:

x² - x + 1 = (x - 1/2)² + 3/4.

From the factorization, we can see that the minimum value of (x - 1/2)² is 0, so the minimum value of the denominator is 3/4.

On the other hand, the numerator of f(x) can take any real value since it is a quadratic expression.

As a result, we can conclude that the range of the function f(x) is all real numbers except 0.

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