Question

Find the domains of definition of the following functions.
$y=\sqrt{\frac{3^x-4^x}{2x^2-x-8}}.$

Answer 1

R.E.F image

Domain of function

$y=\sqrt{\frac{3^x-4^x}{2x^2-x-8}},y=\frac{\sqrt{p\left(x\right)}}{q\left(x\right)}$

$p\left(x\right)\ge 0$

$3^x\ge 4^x$

$(\frac{3}{4}{)}^x\ge 1$

use $a^x\ge (a{)}^0$

$a<1$

$x\le 0$

$(\frac{3}{4}{)}^x\ge ({\frac{3}{4}}^0)$

$x\le 0$

$(-\infty ,0)...\left(i\right)$

$2\left(x\right)>0$

$2x^2-x-8>0$

$(x-a)(x-B)>0$

$a=\frac{1-\sqrt{65}}{4},B=\frac{1+\sqrt{65}}{4}$

$xϵ(-\infty ,\frac{1-\sqrt{65}}{4},)\cup (\frac{1+\sqrt{65}}{4},\infty )...\left(ii\right)$

Using intersection of both domain

$xϵ(-\infty ,\frac{1-\sqrt{65}}{4})$