Question

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

Answer 1

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

Domain of logarithmic function is positive

$\Rightarrow 1-x>0$

$\Rightarrow x<1$ .... $\left(1\right)$

And the number in square root must be non negative

$\Rightarrow x+2\ge 0$

$\Rightarrow x\ge -2$ ..... $\left(2\right)$

And the denominator should not be equal to zero in first term

$\Rightarrow$ ${\log }_{10}(1-x)\ne 0$

$\Rightarrow$ $x\ne 1$ ...... $\left(3\right)$

Combining all equations;

$\Rightarrow$ $-2\le x<1$

Thus, the domain of $f\left(x\right)$ is $x\in [-2,1)$