Question

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

Answer 1

We have $\sqrt{x+2}$ is defined for $x+2\ge 0$ or, $x\ge -2$........(1).

Now $\frac{1}{{\log }_{10}(1-x)}$ is defined for ${\log }_{10}(1-x)\ne 0$ or, $x\ne 1$......(2) and $(1-x)>0$ or, $x<1$.......(3).

Combining (1),(2) and (3) we get the domain of definition for $f\left(x\right)$ and it is

$[-2,0)\cap (0,1)$.