https://file.aiquickdraw.com/mathbot/file/1725385486611zclwkm5f.jpg

The piecewise function \( f(x) \) given in the image is defined as follows:

\[
f(x) =
\begin{cases}
6 & \text{for } x \leq -4 \\
-x + 12 & \text{for } x > 5
\end{cases}
\]

To graph this function, we need to determine key points for each piece of the function:

### For \( x \leq -4 \):
- The function is constant: \( f(x) = 6 \).
- This means for any \( x \leq -4 \), \( f(x) = 6 \).
- A key point would be at \( (-4, 6) \), and all points to the left of \( x = -4 \) will also have \( y = 6 \).

### For \( x > 5 \):
- The function is linear: \( f(x) = -x + 12 \).
- To find key points, we can substitute values of \( x \) greater than 5 into the function.

Let's calculate a few points:
- When \( x = 6 \), \( f(6) = -6 + 12 = 6 \).
- When \( x = 7 \), \( f(7) = -7 + 12 = 5 \).
- When \( x = 8 \), \( f(8) = -8 + 12 = 4 \).

### Summary of Key Points:
1. For \( x \leq -4 \):
   - \( (-4, 6) \)
   - Any other point with \( x < -4 \) will also have \( y = 6 \), such as \( (-5, 6) \), \( (-6, 6) \), etc.

2. For \( x > 5 \):
   - \( (6, 6) \)
   - \( (7, 5) \)
   - \( (8, 4) \)
   - \( (x, -x + 12) \) for any \( x > 5 \)

Would you like a more detailed explanation of any part, or do you have any questions?

Here are 5 related questions you might consider:
1. How do you graph piecewise functions with more than two conditions?
2. What is the significance of the points where the conditions change in a piecewise function?
3. How do you determine the continuity of a piecewise function?
4. Can a piecewise function be differentiable at the points where its definition changes?
5. How would you find the range of the piecewise function given?

**Tip:** When plotting a piecewise function, pay attention to whether the function is defined as closed or open at the boundaries, which affects whether you plot a filled or open circle at the boundary points.

This page covers how to graph the piecewise function f(x) which has a constant value for x <= -4 and a linear function for x > 5. Learn how to determine the key points and visualize the graph with this step-by-step guide.

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