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range of (x^2+x+1)/x

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range x²+x+1x

Solution

f(x)≤ −1 or f(x)≥ 3

Interval Notation

(−∞ , −1]∪ [3, ∞ )

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Function range definition

The set of values of the dependent variable for which a function is defined

Rewrite as

x²+x+1x =y

Multiply both sides by x

x²+x+1=yx

The range is the set of y for which the discriminant is greater or equal to zero

Discriminant x²+x+1=yx: y²−2y−3

y²−2y−3≥ 0 : y≤ −1 or y≥ 3

Check if the range interval endpoints are included

y=−1 : Included

y=3 : Included

Therefore the range is

f(x)≤ −1 or f(x)≥ 3

Solve instead

domain x²+x+1x

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−▭▭	<	7	8	9	÷	`AC`
+▭▭	>	4	5	6	×	☐☐☐
×▭▭	(	1	2	3	−	`x`
▭ \|▭	)	.	0	=	+	`y`

range of (x^2+x+1)/x

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