What is the domain of $sqrt(2x^2+x-6)$ ?

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Answer 1 · 2017-01-03T08:48:24

The domain is $x in ] -oo,-2 ] uu [3/2, oo[$

We factorise

$2x^2+x-6=(2x-3)(x+2)$

Then,

$2x^2+x-6>=0$

$(2x-3)(x+2)>=0$

Let $f(x)=(2x-3)(x+2)$

We have to do a sign chart

$color(white)(aaaa)$ $x$ $color(white)(aaaa)$ $-oo$ $color(white)(aaaa)$ $-2$ $color(white)(aaaaa)$ $3/2$ $color(white)(aaaa)$ $+oo$

$color(white)(aaaa)$ $x+2$ $color(white)(aaaaaa)$ $-$ $color(white)(aaaa)$ $+$ $color(white)(aaaaa)$ $+$

$color(white)(aaaa)$ $2x-3$ $color(white)(aaaaa)$ $-$ $color(white)(aaaa)$ $-$ $color(white)(aaaaa)$ $+$

$color(white)(aaaa)$ $f(x)$ $color(white)(aaaaaaa)$ $+$ $color(white)(aaaa)$ $-$ $color(white)(aaaaa)$ $+$

Therefore,

As $f(x)>=0$

The domain is $x in ] -oo,-2 ] uu [3/2, oo[$

What is the domain of sqrt(2x^2+x-6)sqrt(2x^2+x-6)?