How do you find the domain and range of $\sqrt{25 - x^{2}}$ $sqrt(25-x^2)$ ?

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How do you find the domain and range of $\sqrt{25 - x^{2}}$ $sqrt(25-x^2)$ ?

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Answer 1 · 2017-01-26T11:31:34

The domain is $= [- 5, 5]$ $= [-5,5]$
The range is $= [0, 5]$ $=[0,5]$

Explanation:

What is under the $\sqrt{}$ $sqrt$ sign is $\geq 0$ $>=0$

Therefore,

$25 - x^{2} \geq 0$ $25-x^2>=0$

$(5 + x) (5 - x) \geq 0$ $(5+x)(5-x)>=0$

Let $f (x) = (5 + x) (5 - x)$ $f(x)=(5+x)(5-x)$

We can now build a sign chart

$a a a a$ $color(white)(aaaa)$ $x$ $x$ $a a a a$ $color(white)(aaaa)$ $- \infty$ $-oo$ $a a a a$ $color(white)(aaaa)$ $- 5$ $-5$ $a a a a$ $color(white)(aaaa)$ $5$ $5$ $a a a a$ $color(white)(aaaa)$ $+ \infty$ $+oo$

$a a a a$ $color(white)(aaaa)$ $5 + x$ $5+x$ $a a a a a$ $color(white)(aaaaa)$ $-$ $-$ $a a a a$ $color(white)(aaaa)$ $+$ $+$ $a a a a$ $color(white)(aaaa)$ $+$ $+$

$a a a a$ $color(white)(aaaa)$ $5 - x$ $5-x$ $a a a a a$ $color(white)(aaaaa)$ $+$ $+$ $a a a a$ $color(white)(aaaa)$ $+$ $+$ $a a a a$ $color(white)(aaaa)$ $-$ $-$

$a a a a$ $color(white)(aaaa)$ $f (x)$ $f(x)$ $a a a a a a$ $color(white)(aaaaaa)$ $-$ $-$ $a a a a$ $color(white)(aaaa)$ $+$ $+$ $a a a a$ $color(white)(aaaa)$ $-$ $-$

Therefore,

$f (x) \geq 0$ $f(x)>=0$ when $x \in [- 5, 5]$ $x in [-5, 5]$

The domain of $f (x)$ $f(x)$ is $D_{f (x)} = [- 5, 5]$ $D_f(x)= [-5,5]$

When $x = \pm 5$ $x=+-5$ , $\sqrt{f (x)} = 0$ $sqrt(f(x))=0$

When $x = 0$ $x=0$ , $\sqrt{f (x)} = 5$ $sqrt(f(x))=5$ , this is the max. value

so,
The range is $R = [0, 5]$ $R = [0,5]$

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How do you find the domain and range of $\sqrt{25 - x^{2}}$ $sqrt(25-x^2)$ ?

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How do you find the domain and range of √25−x2sqrt(25-x^2) ?

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How do you find the domain and range of $\sqrt{25 - x^{2}}$ $sqrt(25-x^2)$ ?