Let's break down the tasks step by step based on the given image.

1. Functions and their Domains:

• $f(x) = 9x^2 + 6$, where the domain $M_f = [-\infty, -1]$.
• $g(x) = -\frac{1}{3} \sqrt{x - 6}$, where the domain $M_g = [7, \infty[$.

The task is to find the composition of these functions, along with the associated domains, and also the inverse of another function $h(x)$.

2. Composition of Functions:

You are asked to compute $f \circ g$ and $g \circ f$, then find the domains $M_{f \circ g}$ and $M_{g \circ f}$.

a. Finding $f \circ g$ and its domain:

• $f \circ g(x) = f(g(x))$, which means you substitute $g(x)$ into $f(x)$.
• Given that $g(x) = -\frac{1}{3} \sqrt{x - 6}$, substitute this into $f(x) = 9x^2 + 6$.
• Find the domain where this composition is valid. It will be the intersection of the domains of $f$ and $g$, considering the range of $g(x)$ must fit within the domain of $f(x)$.

b. Finding $g \circ f$ and its domain:

• $g \circ f(x) = g(f(x))$, which means you substitute $f(x)$ into $g(x)$.
• Substitute $f(x) = 9x^2 + 6$ into $g(x) = -\frac{1}{3} \sqrt{x - 6}$.
• Similarly, find the domain $M_{g \circ f}$, ensuring the input of $f(x)$ leads to valid outputs in the domain of $g(x)$.

3. Inverse of $h(x)$:

The function $h(x) = \sqrt{x^2 + 28} - 5$, with $x \in [0, 2]$, is given. You are asked to find the inverse function $h^{-1}(x)$ and its domain $M_{h^{-1}}$.

• To find the inverse, solve $y = \sqrt{x^2 + 28} - 5$ for $x$ in terms of $y$.
• Determine the domain where the inverse function is valid, ensuring you respect the domain restriction of $h(x)$.

I will now work through these calculations and provide the solutions.Here are the results from the calculations:

1. Composition $f \circ g(x)$:

• $f \circ g(x) = 9\left(-\frac{1}{3} \sqrt{x - 6}\right)^2 + 6 = x$.
• The domain $M_{f \circ g}$ is where $g(x)$ is valid within the domain of $f(x)$, which would be $x \in [7, \infty[$.

2. Composition $g \circ f(x)$:

• $g \circ f(x) = -\frac{1}{3} \sqrt{9x^2 + 6 - 6} = -\sqrt{x^2}$.
• This simplifies to $-|x|$, with the domain $M_{g \circ f}$ requiring $9x^2 + 6 \geq 6$, so $x \in [-1, -\infty]$.

3. Inverse of $h(x)$:

• The inverse of $h(x) = \sqrt{x^2 + 28} - 5$ is: $h^{-1}(x) = \pm \sqrt{(x + 5)^2 - 28}$
• Given the domain of $h(x)$ is $[0, 2]$, we choose the positive branch $h^{-1}(x) = \sqrt{(x + 5)^2 - 28}$.
• The domain of the inverse, $M_{h^{-1}}$, depends on the range of $h(x)$, which is the output of $h(x)$. This leads to $x \in [-5, 0]$.

Would you like further details on any step? Here are 5 related questions:

1. How do we determine the range of the composite function $f \circ g$?
2. How does the absolute value affect the result of $g \circ f$?
3. Why do we select the positive root for the inverse of $h(x)$?
4. Can we visualize these function compositions graphically?
5. How can we verify the domain restrictions for these compositions?

Tip: When working with composite functions, always ensure the output of one function is valid within the domain of the next function in the composition.