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Let's go through the problem and solve for \( f(9) \), \( g(-3) \), and \( h(3) \) step by step. The functions are:

\[
f(x) = \left| -13 + \frac{1}{3}x \right|
\]
\[
g(x) = \frac{8 + x^3}{x^3}
\]
\[
h(x) = \sqrt{-2 + 2x}
\]

### 1. Find \( f(9) \)
Substitute \( x = 9 \) into the function \( f(x) \):

\[
f(9) = \left| -13 + \frac{1}{3}(9) \right| = \left| -13 + 3 \right| = \left| -10 \right| = 10
\]

So, \( f(9) = 10 \).

### 2. Find \( g(-3) \)
Substitute \( x = -3 \) into the function \( g(x) \):

\[
g(-3) = \frac{8 + (-3)^3}{(-3)^3} = \frac{8 + (-27)}{-27} = \frac{-19}{-27} = \frac{19}{27}
\]

So, \( g(-3) = \frac{19}{27} \).

### 3. Find \( h(3) \)
Substitute \( x = 3 \) into the function \( h(x) \):

\[
h(3) = \sqrt{-2 + 2(3)} = \sqrt{-2 + 6} = \sqrt{4} = 2
\]

So, \( h(3) = 2 \).

### Final Answers:
- \( f(9) = 10 \)
- \( g(-3) = \frac{19}{27} \)
- \( h(3) = 2 \)

Would you like further details or explanations for any of the steps?

### Related questions:
1. What would \( f(x) \) look like graphically as an absolute value function?
2. How does changing the coefficient in \( g(x) \) affect its graph?
3. Can \( h(x) \) take negative values for any \( x \)?
4. What happens when you substitute non-integer values into these functions?
5. How do transformations of functions (like absolute value and square roots) affect their domains?

#### Tip: When dealing with functions involving absolute values, always check the sign inside before applying the absolute value to simplify calculations!

Find f(9), g(-3), and h(3) based on the functions provided: f(x) = |−13 + (1/3)x|, g(x) = (8 + x^3)/x^3, and h(x) = sqrt(-2 + 2x).

Learn how to evaluate the functions f(x), g(x), and h(x) for specific values. This problem includes key concepts like absolute value, rational expressions, and square roots, suitable for students in Grades 10-12.

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