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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.

Let g(x) be the function defined by g(x) = -x^2 + 6x. Find g(a + h), g(-a), g(√a), a + g(a), and 1/g(a).

Explore how to evaluate the quadratic function g(x) = -x^2 + 6x for various inputs such as a + h, -a, √a, and a. This detailed solution covers the steps for each calculation, suitable for students in Grades 10-12.

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