Question

find the domain of the following functions :

Answer 1

We only have to consider the denominator.

$\sqrt{(0.2)^{x^2+9x+5}-125}$

We know that the denominator of a fraction can't be zero.

$\begin{aligned}&\sqrt{(0.2)^{x^2+9x+5}-125}\neq 0\\\\\implies\ \ &\left(\dfrac{1}{5}\right)^{x^2+9x+5}-125\neq 0\\\\\implies\ \ &(5^{-1})^{x^2+9x+5}\neq 125\\\\\implies\ \ &5^{-(x^2+9x+5)}\neq 5^3\end{aligned}$

Taking the exponents,

$\begin{aligned}&-(x^2+9x+5)\neq 3\\\\\implies\ \ &x^2+9x+5\neq -3\\\\\implies\ \ &x^2+9x+8\neq 0\\\\\implies\ \ &x^2+x+8x+8\neq 0\\\\\implies\ \ &x(x+1)+8(x+1)\neq 0\\\\\implies\ \ &(x+1)(x+8)\neq 0\\\\\implies\ \ &x\neq -1\quad;\quad x\neq -8\end{aligned}$

Hence the domain is,

$\large\boxed{\mathbb{R}-\{-8,\ -1\}}$