What is the range of x^4-9x^2 ?

The range of x^4-9x^2 is f(x)>=-81/4

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range of x^4-9x^2

range

x^{4} - 9 x^{2}

f (x) \geq - \frac{81}{4}

Interval Notation

[- \frac{81}{4}, \infty)

Solution steps

The function range is the combined domain of the inverse functions

Find the inverse function(s) of:

x^{4} - 9 x^{2}

Inverse of

x^{4} - 9 x^{2} : \frac{9 + 81 + 4 x}{2}, - \frac{9 + 81 + 4 x}{2}, \frac{9 - 81 + 4 x}{2}, - \frac{9 - 81 + 4 x}{2}

\frac{9 + 81 + 4 x}{2}, - \frac{9 + 81 + 4 x}{2}, \frac{9 - 81 + 4 x}{2}, - \frac{9 - 81 + 4 x}{2}

Find the domain of each inverse function

Domain of

\frac{9 + 81 + 4 x}{2} : x \geq - \frac{81}{4}

Domain of

- \frac{9 + 81 + 4 x}{2} : x \geq - \frac{81}{4}

Domain of

\frac{9 - 81 + 4 x}{2} : - \frac{81}{4} \leq x \leq 0

Domain of

- \frac{9 - 81 + 4 x}{2} : - \frac{81}{4} \leq x \leq 0

Combine the intervals

f (x) \geq - \frac{81}{4}

f (x) = 9 x - 4

3 x

f (x) = \frac{6}{5} x^{2} + \frac{3}{2}

(11.2, - 2.2), (5.2, - 10.2)

f (x) = - \frac{1}{4} x + 15