Worksheet for evaluating piecewise functions with numerical inputs.

Let's evaluate each piecewise function at the given value of $ x $. We'll go through each problem one by one, determine which condition applies to the given input, and compute the corresponding output.

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1. Evaluate $ f(x) $ at $ x = 8 $

$$
f(x) =
\begin{cases}
-\frac{x}{9} + 1 & ,\ x = -8 \\
-1 & ,\ x = 8 \\
-\frac{x}{11} - 7 & ,\ x \neq -8, 8
\end{cases}
$$

Since $ x = 8 $, we use the second case:
$$
f(8) = -1
$$

✔ Answer: $ \boxed{-1} $

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2. Evaluate $ f(x) $ at $ x = 10 $

$$
f(x) =
\begin{cases}
-9 & ,\ x = -2 \\
14 & ,\ x = 10 \\
2 & ,\ x \neq -2, 10
\end{cases}
$$

Since $ x = 10 $, we use the second case:
$$
f(10) = 14
$$

✔ Answer: $ \boxed{14} $

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3. Evaluate $ f(x) $ at $ x = 8 $

$$
f(x) =
\begin{cases}
-3 & ,\ x = 0 \\
\frac{x}{8} - 10 & ,\ x = 8 \\
\frac{12}{x} + 4 & ,\ x \neq 0, 8
\end{cases}
$$

Since $ x = 8 $, we use the second case:
$$
f(8) = \frac{8}{8} - 10 = 1 - 10 = -9
$$

✔ Answer: $ \boxed{-9} $

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4. Evaluate $ f(x) $ at $ x = 10 $

$$
f(x) =
\begin{cases}
\frac{x}{7} - 8 & ,\ x = -2 \\
7 & ,\ x = -5 \\
(x - 2)^2 & ,\ x \neq -2, -5
\end{cases}
$$

Now, $ x = 10 $, which is not equal to $-2$ or $-5$, so we use the third case:
$$
f(10) = (10 - 2)^2 = 8^2 = 64
$$

✔ Answer: $ \boxed{64} $

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5. Evaluate $ f(x) $ at $ x = -5 $

$$
f(x) =
\begin{cases}
(x - 3)^2 & ,\ x = -8 \\
5 & ,\ x = -5 \\
-\frac{x}{8} - 1 & ,\ x \neq -8, -5
\end{cases}
$$

Since $ x = -5 $, we use the second case:
$$
f(-5) = 5
$$

✔ Answer: $ \boxed{5} $

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6. Evaluate $ f(x) $ at $ x = 7 $

$$
f(x) =
\begin{cases}
-\frac{x}{11} + 15 & ,\ x \geq 10 \\
x^2 + 14 & ,\ x < 10
\end{cases}
$$

Since $ x = 7 $, and $ 7 < 10 $, we use the second case:
$$
f(7) = 7^2 + 14 = 49 + 14 = 63
$$

✔ Answer: $ \boxed{63} $

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✔ Final Answers:

1. $ f(8) = \boxed{-1} $
2. $ f(10) = \boxed{14} $
3. $ f(8) = \boxed{-9} $
4. $ f(10) = \boxed{64} $
5. $ f(-5) = \boxed{5} $
6. $ f(7) = \boxed{63} $

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🔍 Summary of Strategy:

For piecewise functions:
- Identify the value of $ x $ you're evaluating.
- Check which condition the input satisfies.
- Use the corresponding expression for that condition.
- Simplify the result.

This approach ensures accuracy when dealing with multiple rules based on different domains.