Question

Let $f\left(x\right)=\sqrt{x+2}+\frac{1}{{\log }_{10}(1-x)}$. Then find the domain of $f\left(x\right)$.

Answer 1

$f\left(x\right)=\sqrt{x+2}+\frac{1}{{\log }_{10}(1-x)}$.

Now $\sqrt{x+2}$ is defined for $x+2\ge 0\Rightarrow x\ge -2$.....(1).

Again $\frac{1}{{\log }_{10}(1-x)}$ is defined for ${\log }_{10}(1-x)\ne 0$ and $(1-x)>0$.

or, $x\ne 0$ and $x<1$.

So the domain for $f\left(x\right)$ is $[-2,0)\cap (0,1)$.