Understand the given functions.

Step 1: Understand the given functions. We are given two functions: $f(x) = x^2 + 1$ $g(x) = x + 1$

Step 2: Solve part a). a) Find $(f \circ g)(1)$. This means $f(g(1))$. First, evaluate $g(1)$: $g(1) = 1 + 1 = 2$ Next, evaluate $f(2)$: $f(2) = (2)^2 + 1 = 4 + 1 = 5$ So, $(f \circ g)(1) = \boxed{5}$.

Step 3: Solve part b). b) Find $(g \circ f)(1)$. This means $g(f(1))$. First, evaluate $f(1)$: $f(1) = (1)^2 + 1 = 1 + 1 = 2$ Next, evaluate $g(2)$: $g(2) = 2 + 1 = 3$ So, $(g \circ f)(1) = \boxed{3}$.

Step 4: Solve part c). c) Find $(f \circ g)(x)$. This means $f(g(x))$. Substitute $g(x)$ into $f(x)$: $f(g(x)) = f(x+1)$ Now, substitute $(x+1)$ into the expression for $f(x)$: $f(x+1) = (x+1)^2 + 1$ Expand the expression: $f(x+1) = (x^2 + 2x + 1) + 1$ $f(x+1) = x^2 + 2x + 2$ So, $(f \circ g)(x) = \boxed{x^2 + 2x + 2}$.

Step 5: Solve part d). d) Find $(g \circ f)(x)$. This means $g(f(x))$. Substitute $f(x)$ into $g(x)$: $g(f(x)) = g(x^2+1)$ Now, substitute $(x^2+1)$ into the expression for $g(x)$: $g(x^2+1) = (x^2+1) + 1$ $g(x^2+1) = x^2 + 2$ So, $(g \circ f)(x) = \boxed{x^2 + 2}$.

Step 6: Solve part e). e) Find $(f \circ g)(a^2)$. This means $f(g(a^2))$. First, evaluate $g(a^2)$: $g(a^2) = a^2 + 1$ Next, evaluate $f(a^2+1)$: $f(a^2+1) = (a^2+1)^2 + 1$ Expand the expression: $f(a^2+1) = (a^4 + 2a^2 + 1) + 1$ $f(a^2+1) = a^4 + 2a^2 + 2$ So, $(f \circ g)(a^2) = \boxed{a^4 + 2a^2 + 2}$.

Step 7: Solve part f). f) Find $(g \circ f)(\sqrt{a})$. This means $g(f(\sqrt{a}))$. First, evaluate $f(\sqrt{a})$: $f(\sqrt{a}) = (\sqrt{a})^2 + 1 = a + 1$ Next, evaluate $g(a+1)$: $g(a+1) = (a+1) + 1 = a + 2$ So, $(g \circ f)(\sqrt{a}) = \boxed{a + 2}$.