What is the range of f(x)=x^4-29x^2+100 ?

The range of f(x)=x^4-29x^2+100 is f(x)>=-441/4

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range of f(x)=x^4-29x^2+100

range

f (x) = x^{4} - 29 x^{2} + 100

f (x) \geq - \frac{441}{4}

Interval Notation

[- \frac{441}{4}, \infty)

Solution steps

The function range is the combined domain of the inverse functions

Find the inverse function(s) of:

x^{4} - 29 x^{2} + 100

Inverse of

x^{4} - 29 x^{2} + 100 : \frac{29 + 4 x + 441}{2}, - \frac{29 + 4 x + 441}{2}, \frac{29 - 4 x + 441}{2}, - \frac{29 - 4 x + 441}{2}

\frac{29 + 4 x + 441}{2}, - \frac{29 + 4 x + 441}{2}, \frac{29 - 4 x + 441}{2}, - \frac{29 - 4 x + 441}{2}

Find the domain of each inverse function

Domain of

\frac{29 + 4 x + 441}{2} : x \geq - \frac{441}{4}

Domain of

- \frac{29 + 4 x + 441}{2} : x \geq - \frac{441}{4}

Domain of

\frac{29 - 4 x + 441}{2} : - \frac{441}{4} \leq x \leq 100

Domain of

- \frac{29 - 4 x + 441}{2} : - \frac{441}{4} \leq x \leq 100

Combine the intervals

f (x) \geq - \frac{441}{4}

What is the range of f(x)=x^4-29x^2+100 ?
The range of f(x)=x^4-29x^2+100 is f(x)>=-441/4