The problem in the image asks you to find several values based on the function $g(x) = -x^2 + 6x$. The specific values to calculate are:

1. $g(a + h)$
2. $g(-a)$
3. $g(\sqrt{a})$
4. $a + g(a)$
5. $\frac{1}{g(a)}$

Let's go step-by-step through each part:

1. $g(a + h)$

To find $g(a + h)$, substitute $a + h$ for $x$ in the function $g(x) = -x^2 + 6x$:

$g(a + h) = -(a + h)^2 + 6(a + h)$ Now expand: $g(a + h) = -(a^2 + 2ah + h^2) + 6a + 6h$ $g(a + h) = -a^2 - 2ah - h^2 + 6a + 6h$

2. $g(-a)$

Substitute $-a$ for $x$:

$g(-a) = -(-a)^2 + 6(-a)$ $g(-a) = -a^2 - 6a$

3. $g(\sqrt{a})$

Substitute $\sqrt{a}$ for $x$:

$g(\sqrt{a}) = -(\sqrt{a})^2 + 6\sqrt{a}$ $g(\sqrt{a}) = -a + 6\sqrt{a}$

4. $a + g(a)$

First, find $g(a)$ by substituting $a$ for $x$:

$g(a) = -a^2 + 6a$ Now add $a$ to this result: $a + g(a) = a + (-a^2 + 6a) = -a^2 + 7a$

5. $\frac{1}{g(a)}$

Using the expression for $g(a)$ from the previous step: $\frac{1}{g(a)} = \frac{1}{-a^2 + 6a}$

These are the general expressions. To get the final numerical answers, you would need the specific value of $a$ and $h$.

Let me know if you want further clarifications or if you have a specific value for $a$ and $h$ to proceed with the final answers.

Here are some related questions to extend the concept:

1. How do you find the derivative of $g(x)$?
2. What is the vertex of the quadratic function $g(x) = -x^2 + 6x$?
3. How does $g(a + h)$ relate to the concept of limits and derivatives?
4. How would you solve for $g(x) = 0$?
5. Can you rewrite $g(x)$ in vertex form?

Tip: Always simplify each expression step-by-step and look for patterns when evaluating functions with variables.