# How do you find the domain of f(x) = arc cot( (x−1) / ((1−x)^0.5) ) + log_(1/3) (2x + 21)?

Domain: $\left(- 10 \frac{1}{2} , 1\right)$

#### Explanation:

The given equation is
$f \left(x\right) = {\cot}^{-} 1 \left(\frac{x - 1}{1 - x} ^ 0.5\right) + {\log}_{\frac{1}{3}} \left(2 x + 21\right)$

For the term ${\cot}^{-} 1 \left(\frac{x - 1}{1 - x} ^ 0.5\right)$

Domain: $\left(- \infty , 1\right)$

For the term ${\log}_{\frac{1}{3}} \left(2 x + 21\right)$

Domain: $\left(- 10 \frac{1}{2} , + \infty\right)$

Therefore the domain for the entire equation is the intersection which is

Domain: $\left(- 10 \frac{1}{2} , 1\right)$

God bless....I hope the explanation is useful.