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!