What is the range of (2x^4-x^2+1)/(x^2-4) ?

The range of (2x^4-x^2+1)/(x^2-4) is f(x) = 2sqrt(58)+15

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range of (2x^4-x^2+1)/(x^2-4)

range

f (x) \leq - 258 + 15 or f (x) \geq 258 + 15

Interval Notation

(- \infty, - 258 + 15] \cup [258 + 15, \infty)

Solution steps

The function range is the combined domain of the inverse functions

Find the inverse function(s) of:

\frac{2 x ^{4} - x ^{2} + 1}{x ^{2} - 4}

Inverse of

\frac{2 x ^{4} - x ^{2} + 1}{x ^{2} - 4} : \frac{1 + x + x ^{2} - 30 x - 7}{2}, - \frac{1 + x + x ^{2} - 30 x - 7}{2}, \frac{1 + x - x ^{2} - 30 x - 7}{2}, - \frac{1 + x - x ^{2} - 30 x - 7}{2}

\frac{1 + x + x ^{2} - 30 x - 7}{2}, - \frac{1 + x + x ^{2} - 30 x - 7}{2}, \frac{1 + x - x ^{2} - 30 x - 7}{2}, - \frac{1 + x - x ^{2} - 30 x - 7}{2}

Find the domain of each inverse function

Domain of

\frac{1 + x + x ^{2} - 30 x - 7}{2} : x \leq - 258 + 15 or x \geq 258 + 15

Domain of

- \frac{1 + x + x ^{2} - 30 x - 7}{2} : x \leq - 258 + 15 or x \geq 258 + 15

Domain of

\frac{1 + x - x ^{2} - 30 x - 7}{2} : - \frac{1}{4} \leq x \leq - 258 + 15 or x \geq 258 + 15

Domain of

- \frac{1 + x - x ^{2} - 30 x - 7}{2} : - \frac{1}{4} \leq x \leq - 258 + 15 or x \geq 258 + 15

Combine the intervals

f (x) \leq - 258 + 15 or f (x) \geq 258 + 15

What is the range of (2x^4-x^2+1)/(x^2-4) ?
The range of (2x^4-x^2+1)/(x^2-4) is f(x)<=-2sqrt(58)+15\lor f(x)>= 2sqrt(58)+15