How do you find the domain and range of $sqrt [x/(x-6)]$ ?

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Answer 1 · 2017-04-12T12:01:57

The domain is $x in (-oo,0]uu(6,+oo)$
The range is $[0,1)uu(1,+oo)$

To find the domain, what's under the square root sign is $>=0$

So,

$x/(x-6)>=0$

Let $p(x)=x/(x-6)$

We need to build a sign chart

$color(white)(aaaa)$ $x$ $color(white)(aaaa)$ $-oo$ $color(white)(aaaa)$ $0$ $color(white)(aaaaaaaa)$ $6$ $color(white)(aaaaaaaa)$ $+oo$

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

$color(white)(aaaa)$ $x-6$ $color(white)(aaaa)$ $-$ $color(white)(aaaa)$ $-$ $color(white)(aaaaa)$ $||$ $color(white)(aaaa)$ $+$

$color(white)(aaaa)$ $p(x)$ $color(white)(aaaaa)$ $+$ $color(white)(aaaa)$ $-$ $color(white)(aaaaa)$ $||$ $color(white)(aaaa)$ $+$

Therefore,

$p(x)>=0$ when $x in (-oo,0]uu(6,+oo)$

The domain is $x in (-oo,0]uu(6,+oo)$

$lim_(x->+-oo)p(x)=lim_(x->+-oo)x/x=1$

When $x=0$ , $p(x)=0$

$lim_(x->6^+)p(x)=+oo$

So,

the range is $[0,1)uu(1,+oo)$

graph{sqrt(x/(x-6)) [-25.67, 25.65, -12.83, 12.84]}

How do you find the domain and range of sqrt [x/(x-6)]sqrt [x/(x-6)]?