Here is a step-by-step solution to the questions based on the provided image:

6. On which interval of $x$ is the function decreasing?

• The function decreases when its slope is negative.
• From the graph, the function is decreasing on the interval: $x \in (2, 6)$.

7. Write $g(x)$ in terms of $f(x)$:

• From the graph, we observe that $g(x)$ is a reflection and shift of $f(x)$.
• Specifically: $g(x) = -f(x+2)$.
  (Explanation: The function $g(x)$ is flipped vertically and shifted left by 2 units compared to $f(x)$.)

8. Is $f(x) = |x|$ odd, even, or neither?

• A function is even if $f(-x) = f(x)$ and odd if $f(-x) = -f(x)$.
• For $f(x) = |x|$: $f(-x) = |x| = f(x)$.
• Therefore, $f(x) = |x|$ is even.

9. Find the inverse of $f(x) = \frac{x}{2} + 3$:

1. Let $y = f(x) = \frac{x}{2} + 3$.
2. Solve for $x$ in terms of $y$: $y - 3 = \frac{x}{2}$ $x = 2(y - 3)$.
3. Replace $y$ with $x$ for the inverse function: $f^{-1}(x) = 2(x - 3)$.

Inverse: $f^{-1}(x) = 2x - 6$.

10. Are $f(x) = 3x + x^2$ and $g(x) = \sqrt{x + 1}$ inverses?

• To check if $f(x)$ and $g(x)$ are inverses, verify if: $f(g(x)) = x$ and $g(f(x)) = x$.

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

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

Conclusion: $f(x)$ and $g(x)$ are not inverses of each other.

Let me know if you need further clarifications or additional explanations. Here are related questions you might consider:

1. How can you determine whether a function is increasing or decreasing algebraically?
2. What are some other transformations of functions you can apply using $f(x)$?
3. Can an odd function also be invertible? Why or why not?
4. How do you graphically verify the inverse of a function?
5. What is the significance of checking $f(g(x)) = g(f(x)) = x$ for inverses?

Tip: When finding inverses, always verify the domain and range restrictions to ensure the functions are well-defined.