f(x)= 
⎩
⎨
⎧
​
  
−(x+7) 
2
 +6
−4
−2x−4
​
  
 for x≤−4 
1
1
​
 
 for −4<x≤−1 
1
1
​
 
 for x>−1 
1
1
​
 
​
 
Find , f, of, minus, 4
Find f(−4)

Given the piecewise function:

\[
f(x) = 
\begin{cases} 
-(x + 7)^2 + 6 & \text{for } x \leq -4 \\
-4 & \text{for } -4 < x \leq -1 \\
-2x - 4 & \text{for } x > -1 
\end{cases}
\]

We are tasked with finding \( f(-4) \).

Since \( -4 \leq -4 \), we use the first piece of the function, which is:

\[
f(x) = -(x + 7)^2 + 6
\]

Substitute \( x = -4 \):

\[
f(-4) = -((-4 + 7)^2) + 6
\]
\[
f(-4) = -(3^2) + 6
\]
\[
f(-4) = -9 + 6
\]
\[
f(-4) = -3
\]

So, \( f(-4) = -3 \).

---

Do you want more details or have any questions?

Here are 5 related questions to expand on this concept:

1. What is \( f(-5) \) for the given function?
2. How do we calculate \( f(-1) \)?
3. Can you find \( f(0) \) for this piecewise function?
4. What does the graph of this piecewise function look like?
5. How does the first piece of the function behave as \( x \to -\infty \)?

**Tip:** Always ensure that you're using the correct piece of the function by checking which interval the input falls into.

Find f(−4) for the piecewise function f(x) = -(x + 7)^2 + 6 for x ≤ −4, -4 for −4 < x ≤ −1, and -2x - 4 for x > −1.

Learn how to evaluate f(-4) for a given piecewise function with multiple cases, including a quadratic expression. This problem highlights function evaluation techniques in piecewise functions, suitable for students in Grades 10-12.

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