Let’s solve the given problem step by step. The problem asks us to find the composition of functions $h(x) = (f \circ g)(x)$ and determine the domain of $h(x)$. The functions provided are:

1. $f(x) = x^2 + 1$
2. $g(x) = \sqrt{3x + 6}$

Solution:

a. Find $h(x) = (f \circ g)(x)$:

The composition of $f$ and $g$, $h(x) = (f \circ g)(x)$, is defined as: $h(x) = f(g(x))$

1. Start by substituting $g(x)$ into $f(x)$: $f(g(x)) = f\left(\sqrt{3x + 6}\right)$

2. The function $f(x) = x^2 + 1$ takes the input $x$ and squares it, then adds 1. So: $f\left(\sqrt{3x + 6}\right) = \left(\sqrt{3x + 6}\right)^2 + 1$

3. Simplify: $f\left(\sqrt{3x + 6}\right) = 3x + 6 + 1$

4. Final expression for $h(x)$: $h(x) = 3x + 7$

b. Find the domain of $h(x)$:

To determine the domain of $h(x)$, we need to consider the domain of the inner function $g(x) = \sqrt{3x + 6}$.

1. The square root function $\sqrt{3x + 6}$ is defined when its argument is non-negative: $3x + 6 \geq 0$

2. Solve the inequality: $3x \geq -6 \quad \Rightarrow \quad x \geq -2$

Thus, the domain of $h(x)$ is: $x \geq -2$

Final Answers:

1. $h(x) = 3x + 7$
2. Domain of $h(x)$: $x \geq -2$

Would you like more detailed steps or further explanations? Here are five related questions for practice:

1. How do you find the composition of two functions $f$ and $g$ when both are quadratic?
2. If $g(x) = \sqrt{3x + 6}$, what happens to the domain of $g(x)$ when $f(x)$ has a restriction (e.g., division by zero)?
3. What is the range of $h(x) = 3x + 7$?
4. How does the domain of a composed function depend on the domains of $f(x)$ and $g(x)$?
5. What are the properties of square root functions used in composition problems?

Tip:

When finding the domain of a composition of functions, always check constraints from both the inner and outer functions!