How do you find the domain of the function f $(x)=sqrt(x^3-64 x)$ ?

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How do you find the domain of the function f $(x)=sqrt(x^3-64 x)$ ?

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Answer 1 · 2017-05-19T09:58:11

The domain of $f(x)$ is $x in [-8,0] uu [8,+oo)$

Explanation:

What's under the square root sign is $>=0$

So,

$x^3-64x>=0$

$x^3-64x=x(x^2-64)=x(x+8)(x-8)$

Let $g(x)=x(x+8)(x-8)>=0$

We build a sign chart

$color(white)(aaaa)$ $x$ $color(white)(aaaa)$ $-oo$ $color(white)(aaaa)$ $-8$ $color(white)(aaaa)$ $0$ $color(white)(aaaaa)$ $+8$ $color(white)(aaaaa)$ $+oo$

$color(white)(aaaa)$ $x+8$ $color(white)(aaaaa)$ $-$ $color(white)(aaaa)$ $+$ $color(white)(aaaa)$ $+$ $color(white)(aaaaa)$ $+$

$color(white)(aaaa)$ $x$ $color(white)(aaaaaaaa)$ $-$ $color(white)(aaaa)$ $-$ $color(white)(aaaa)$ $+$ $color(white)(aaaaa)$ $+$

$color(white)(aaaa)$ $x-8$ $color(white)(aaaaa)$ $-$ $color(white)(aaaa)$ $-$ $color(white)(aaaa)$ $-$ $color(white)(aaaaa)$ $+$

$color(white)(aaaa)$ $g(x)$ $color(white)(aaaaaa)$ $-$ $color(white)(aaaa)$ $+$ $color(white)(aaaa)$ $-$ $color(white)(aaaaa)$ $+$

Therefore,

$g(x)>=0$ when $x in [-8,0] uu [8,+oo)$ graph{sqrt(x^3-64x) [-27.54, 30.2, -4.14, 24.74]}

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How do you find the domain of the function f $(x)=sqrt(x^3-64 x)$ ?

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How do you find the domain of the function f $(x)=sqrt(x^3-64 x)$ ?