Compose the following function and state the domain of the composed function. f(x)=1x−1 g(x)=1−x2 a)...

To find g(f(−2)), substitute -2 for x in the function ${\displaystyle f\left(x\right)=\frac{1}{x-1}}$ and then ${\displaystyle -\frac{1}{3}}$ for x in the function ${\displaystyle g\left(x\right)=\sqrt{1-x^2}}$, and simplify.
${\displaystyle g\left(f(-2)\right)=g\left(\frac{1}{-2-1}\right)}$
${\displaystyle =\sqrt{1-{(-\frac{1}{3})}^2}}$
${\displaystyle =\sqrt{1-\frac{1}{9}}}$
${\displaystyle =\sqrt{\frac{9-1}{9}}}$
${\displaystyle =\frac{\sqrt{8}}{3}}$
${\displaystyle =\frac{2\sqrt{2}}{3}}$
Therefore, ${\displaystyle g\left(f(-2)\right)=\frac{2\sqrt{2}}{3}}$
To obtain the composite function ${\displaystyle (g\circ f)\left(x\right)}$, substitute ${\displaystyle f\left(x\right)=\frac{1}{x-1}}$ and ${\displaystyle g\left(x\right)=\sqrt{1-x^2}}$ in the definition of composite function ${\displaystyle (g\circ f)\left(x\right)=g\left[f\left(x\right)\right]}$, and simplify.
${\displaystyle (g\circ f)\left(x\right)=g\left[f\left(x\right)\right]}$
${\displaystyle =g\left|\frac{1}{x-1}\right|}$
${\displaystyle =\sqrt{1-{\left(\frac{1}{x-1}\right)}^2}}$
${\displaystyle =\sqrt{\frac{{(x-1)}^2-1}{{(x-1)}^2}}}$
${\displaystyle =(\frac{\sqrt{x^2-2x+1-1}}{x-1}}$
${\displaystyle =\frac{\sqrt{x(x-2)}}{x-1}}$
Therefore, ${\displaystyle g\left[f\left(x\right)\right]=\frac{\sqrt{x(x-2)}}{x-1}}$
To determine the domain of the composite function ${\displaystyle (g\circ f)\left(x\right)}$, use the definition of the composite function ${\displaystyle g\left[f\left(x\right)\right]=\frac{\sqrt{x(x-2)}}{x-1}}$ and identify the values of x, for which the denominator is not equal to zero and radicand is positive.
${\displaystyle x\le 0}$
${\displaystyle x-2\ge 0}$
${\displaystyle x\ge 2}$
${\displaystyle x-1\ne 0}$
${\displaystyle x\ne 1}$
Hence, the domain of the composite function ${\displaystyle g\left[f\left(x\right)\right]=\frac{\sqrt{x(x-2)}}{x-1}}$ is