Verify some formulas for the function g(x) = (4-x^2)^(1/2)

Let $g(x) \sqrt{4-x^2}$ for $|x| \leq 2$ .

Proof.
$g(-x) = \sqrt{4-(-x^2)} = \sqrt{4-x^2} = g(x)$ . Valid for $|x| \leq 2. \qquad \blacksquare$
Proof.
$g(2y) = \sqrt{4-(2y)^2} = \sqrt{4-4y^2} = 2 \sqrt{1-y^2}$ . Valid for $|y| \leq 1. \qquad \blacksquare$ .
Proof.
$\displaystyle{ g(1/t) = \sqrt{4 - \left( \frac{1}{t} \right)^2} = \sqrt{\frac{1}{t^2}(4t^2 - 1)} = \frac{\sqrt{4t^2-1}}{\sqrt{t^2}} = \frac{\sqrt{4t^2-1}}{|t|}}$ . Valid for $|t| \geq \frac{1}{2}. \qquad \blacksquare$
Proof.
$g(a-2) = \sqrt{4-(a-2)^2} = \sqrt{4-a^2 + 4a - 4} = \sqrt{4a - a^2}$ . Valid for $0 \leq a \leq 4. \qquad \blacksquare$
Proof.
$\displaystyle{g \left( \frac{s}{2} \right) = \sqrt{4-\frac{s^2}{4}} = \frac{1}{2} \sqrt{16-s^2}}$ . Valid for $|s| \leq 4. \qquad \blacksquare$
Proof.
$\displaystyle{\frac{1}{2+g(x)} = \frac{1}{2+\sqrt{4-x^2}} = \frac{2-\sqrt{4-x^2}}{4-4+x^2} = \frac{2-g(x)}{x^2}}$ . Valid for $0 < |x| \leq 2. \qquad \blacksquare$