Let's solve these questions step by step:

1. Given Functions:

$f(x) = \frac{x}{x - 1}$ and $g(x) = \frac{1}{\sqrt{1 + x^2}}$

We are asked to find the following values:

a) $(f + g)(2)$:

$(f + g)(2) = f(2) + g(2)$

• $f(2) = \frac{2}{2 - 1} = 2$
• $g(2) = \frac{1}{\sqrt{1 + 2^2}} = \frac{1}{\sqrt{5}}$

Thus: $(f + g)(2) = 2 + \frac{1}{\sqrt{5}}$

b) $(f \cdot g)(0)$:

$(f \cdot g)(0) = f(0) \cdot g(0)$

• $f(0) = \frac{0}{0 - 1} = 0$
• $g(0) = \frac{1}{\sqrt{1 + 0^2}} = 1$

Thus: $(f \cdot g)(0) = 0 \cdot 1 = 0$

c) $\left( \frac{g}{f} \right)(3)$:

$\left( \frac{g}{f} \right)(3) = \frac{g(3)}{f(3)}$

• $f(3) = \frac{3}{3 - 1} = \frac{3}{2}$
• $g(3) = \frac{1}{\sqrt{1 + 3^2}} = \frac{1}{\sqrt{10}}$

Thus: $\left( \frac{g}{f} \right)(3) = \frac{\frac{1}{\sqrt{10}}}{\frac{3}{2}} = \frac{2}{3\sqrt{10}}$

d) $(f \circ g)(0)$:

$(f \circ g)(0) = f(g(0))$

• $g(0) = 1$
• $f(1) = \frac{1}{1 - 1}$, which is undefined.

Thus, $(f \circ g)(0)$ is undefined.

e) $(f \cdot g)(\sqrt{8})$:

$(f \cdot g)(\sqrt{8}) = f(\sqrt{8}) \cdot g(\sqrt{8})$

• $f(\sqrt{8}) = \frac{\sqrt{8}}{\sqrt{8} - 1} = \frac{2\sqrt{2}}{2\sqrt{2} - 1}$
• $g(\sqrt{8}) = \frac{1}{\sqrt{1 + (\sqrt{8})^2}} = \frac{1}{\sqrt{9}} = \frac{1}{3}$

Thus: $(f \cdot g)(\sqrt{8}) = \frac{2\sqrt{2}}{2\sqrt{2} - 1} \cdot \frac{1}{3}$

f) $(g \circ f)(0)$:

$(g \circ f)(0) = g(f(0))$

• $f(0) = 0$
• $g(0) = 1$

Thus: $(g \circ f)(0) = g(0) = 1$

2. Given Functions:

$f(x) = x^2 + x$ and $g(x) = \frac{2}{x + 3}$

We are asked to find the following values:

a) $(f - g)(2)$:

$(f - g)(2) = f(2) - g(2)$

• $f(2) = 2^2 + 2 = 6$
• $g(2) = \frac{2}{2 + 3} = \frac{2}{5}$

Thus: $(f - g)(2) = 6 - \frac{2}{5} = \frac{30}{5} - \frac{2}{5} = \frac{28}{5}$

b) $\left( \frac{f}{g} \right)(1)$:

$\left( \frac{f}{g} \right)(1) = \frac{f(1)}{g(1)}$

• $f(1) = 1^2 + 1 = 2$
• $g(1) = \frac{2}{1 + 3} = \frac{2}{4} = \frac{1}{2}$

Thus: $\left( \frac{f}{g} \right)(1) = \frac{2}{\frac{1}{2}} = 4$

c) $g^2(3)$:

$g^2(3) = \left( g(3) \right)^2$

• $g(3) = \frac{2}{3 + 3} = \frac{2}{6} = \frac{1}{3}$

Thus: $g^2(3) = \left( \frac{1}{3} \right)^2 = \frac{1}{9}$

d) $(f \circ g)(1)$:

$(f \circ g)(1) = f(g(1))$

• $g(1) = \frac{1}{2}$
• $f\left( \frac{1}{2} \right) = \left( \frac{1}{2} \right)^2 + \frac{1}{2} = \frac{1}{4} + \frac{1}{2} = \frac{3}{4}$

Thus: $(f \circ g)(1) = \frac{3}{4}$

e) $(g \circ f)(1)$:

$(g \circ f)(1) = g(f(1))$

• $f(1) = 2$
• $g(2) = \frac{2}{5}$

Thus: $(g \circ f)(1) = \frac{2}{5}$

Let me know if you'd like further details or explanations.
Here are five questions for you to consider:

1. How can we determine if a composition of functions is defined or undefined?
2. What is the significance of finding the domain of the composed functions?
3. Can the values of functions change if they are composed in a different order?
4. How do we handle undefined results in function evaluation?
5. How does the behavior of functions change when their domains are restricted?

Tip: Always check the domains of both functions before performing compositions to avoid undefined operations.