The problem asks to evaluate the following compositions of functions $f(x) = 1 + x^2$ and $g(x) = \sqrt{x - 1}$:

a. $f \circ g$

b. $g \circ f$

c. $(g \circ f)(2)$

d. $(f \circ f)(2)$

Let's solve them step by step:

a. $f \circ g$

This means $(f \circ g)(x) = f(g(x))$. Substituting $g(x) = \sqrt{x - 1}$ into $f(x)$:

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

Thus, $(f \circ g)(x) = x$.

b. $g \circ f$

This means $(g \circ f)(x) = g(f(x))$. Substituting $f(x) = 1 + x^2$ into $g(x)$:

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

Thus, $(g \circ f)(x) = |x|$.

c. $(g \circ f)(2)$

Now, we compute $(g \circ f)(2)$:

$f(2) = 1 + 2^2 = 1 + 4 = 5$ $g(5) = \sqrt{5 - 1} = \sqrt{4} = 2$

Thus, $(g \circ f)(2) = 2$.

d. $(f \circ f)(2)$

Finally, we compute $(f \circ f)(2)$:

$f(2) = 1 + 2^2 = 1 + 4 = 5$ $f(f(2)) = f(5) = 1 + 5^2 = 1 + 25 = 26$

Thus, $(f \circ f)(2) = 26$.

Final answers:

a. $(f \circ g)(x) = x$

b. $(g \circ f)(x) = |x|$

c. $(g \circ f)(2) = 2$

d. $(f \circ f)(2) = 26$

Would you like further details on any part of the solution?

Here are some related questions:

1. What is the general process for function composition?
2. How does the domain of a composed function change?
3. What happens when you compose a function with its inverse?
4. Can you always compose two functions?
5. How do compositions apply in real-world problems?

Tip: When working with compositions, it's crucial to respect the domain constraints of both functions.