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!