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The piecewise function given is:

\[
f(x) =
\begin{cases} 
2x + 8 & \text{if } x \leq -2 \\
x^2 - 3 & \text{if } -2 < x \leq 3 \\
\sqrt{x + 3} & \text{if } x > 3 
\end{cases}
\]

You are asked to evaluate \( f(3) - 4f(-3) \).

Let's first break this down step by step:

### 1. Find \( f(3) \)
Since \( 3 \) falls in the second condition, \( -2 < x \leq 3 \), we use the expression \( f(x) = x^2 - 3 \):
\[
f(3) = 3^2 - 3 = 9 - 3 = 6
\]

### 2. Find \( f(-3) \)
Since \( -3 \) falls in the first condition, \( x \leq -2 \), we use the expression \( f(x) = 2x + 8 \):
\[
f(-3) = 2(-3) + 8 = -6 + 8 = 2
\]

### 3. Calculate \( f(3) - 4f(-3) \)
Now, substitute the values:
\[
f(3) - 4f(-3) = 6 - 4(2) = 6 - 8 = -2
\]

Thus, the result is \( -2 \).

Let me know if you would like more details on any part of this problem. Here are five related questions for further practice:

1. Evaluate \( f(-4) + f(2) \) using the same function.
2. What is \( f(0) \) for the given piecewise function?
3. Find \( f(4) \), using the third condition of the function.
4. Determine the value of \( f(-2) \) from the piecewise function.
5. Evaluate \( 2f(1) + f(-1) \) using the function.

**Tip:** For piecewise functions, always identify which condition the input falls into before applying the correct equation.

Evaluate f(3) - 4f(-3) for the piecewise function f(x).

This page provides a step-by-step guide to evaluating the piecewise function f(x) at the points f(3) and f(-3), including the solution f(3) - 4f(-3). This example covers using definitions of piecewise functions including linear and quadratic expressions and the calculation of function values at specific points.

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