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Let's break down the given piecewise functions \( f(x) \) and \( g(x) \) and then solve the required values.

### Function \( f(x) \):
- \( f(x) = x^2 + 5 \), when \( x \leq 2 \)
- \( f(x) = -3x \), when \( x > 2 \)

### Function \( g(x) \):
- \( g(x) = 5x + 7 \), when \( x < 0 \)
- \( g(x) = (x - 1)^2 + 2 \), when \( 0 \leq x < 4 \)
- \( g(x) = -x \), when \( x \geq 4 \)

#### 1. Finding \( f(5) \):
Since \( 5 > 2 \), we use the second piece of \( f(x) \):
\[
f(5) = -3(5) = -15
\]

#### 2. Finding \( g(8) \):
Since \( 8 \geq 4 \), we use the third piece of \( g(x) \):
\[
g(8) = -8
\]

#### 3. Finding \( 2f(5) - g(8) \):
First, we compute \( 2f(5) \):
\[
2f(5) = 2(-15) = -30
\]
Now, subtract \( g(8) \) from \( 2f(5) \):
\[
2f(5) - g(8) = -30 - (-8) = -30 + 8 = -22
\]

### Final Answers:
- \( f(5) = -15 \)
- \( g(8) = -8 \)
- \( 2f(5) - g(8) = -22 \)

Would you like further details or clarification on any step?

Here are 5 questions that build on this:
1. How would the value of \( f(x) \) change if \( x \) were a negative number?
2. What would \( g(0) \) be based on the piecewise definition of \( g(x) \)?
3. How do piecewise functions model different behaviors of real-world systems?
4. How does the domain restriction affect which part of the function is applied?
5. What is the difference between evaluating a continuous function versus a piecewise function?

**Tip:** Always check the domain before plugging in values in a piecewise function to ensure you're using the correct expression!

Find the values of f(5), g(8), and 2f(5) - g(8) given the piecewise functions f(x) and g(x).

Learn how to evaluate the piecewise functions f(x) and g(x) at specific values. This example covers function definitions with different expressions based on domain restrictions, suitable for students in grades 9-12.

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