How to find domain for $f(x) = sqrt(25 - x^2)$ ?

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Answer 1 · 2015-09-29T09:59:42

$[-5,5]$

So that the answer won't be imaginary, $25-x^2$ must be greater than or equal to $0$ .

${x|25-x^2>=0}$
${x|25-x^2-25>=0-25}$
${x|-x^2>=-25}$
${x|(-1)(-x^2)<=(-1)(-25)}$
${x|x^2<=25}$

This will then be separated to two cases:
If $x$ is positive:
${x|x^2<=25}$
${x|x<=5}$
$color(blue)((-oo,5])$

If $x$ is negative:
${x|x^2<=25}$
${x|x>=-5}$
$color(red)([-5,oo))$

We will then get the intersection of the two intervals:
$color(blue)((-oo,5])nncolor(red)([-5,oo))$
$color(magenta)([-5,5]$

How to find domain for f(x) = sqrt(25 - x^2)f(x) = sqrt(25 - x^2)?