Question

Find the domains of definition of the following functions.
$y=\frac{\sqrt{6x-x^2-5}}{5^{x-2}-1}.$

Answer 1

$y=\frac{\sqrt{6x-x^2-5}}{5^{x-2}-1},\frac{\sqrt{p\left(x\right)}}{Q\left(x\right)}$

Domain $\Rightarrow p\left(x\right)\ge 0$ and $Q\left(x\right)\ne 0$

$6x-x^2-5\ge 0$

multiply -1 both side

$x^2-6x+5\le 0$

$(x-5)(x-1)\le 0$

$xϵ[1,5]...\left(i\right)$

and $5^{x-2}-1\ne 0$

$5^{x-2}\ne 1$

or

$5^{x-2}\ne 5^0$

$x-2\ne 0$

$x\ne 2$

$xϵ(-\infty ,\infty )-\left[2\right]...\left(ii\right)$

taking intersection of both values of function

Domain $=xϵ[1,5]-\left\{2\right\}$