https://file.aiquickdraw.com/mathbot/file/1726154340004gfb9at4y.jpg

To analyze and sketch the graph of the given piecewise function \( f(x) \), let's break it down according to the intervals provided:

\[
f(x) = 
\begin{cases} 
x + 4 & \text{if } x < -1 \\
1 & \text{if } 1 \leq x \leq 4 \\
0.5x - 1 & \text{if } x > 4 
\end{cases}
\]

### Step 1: Analyze each piece of the function

1. **For \( x < -1 \), the function is \( f(x) = x + 4 \):**
   - This is a straight line with slope 1 and y-intercept 4.
   - The formula applies up to \( x = -1 \), but does not include \( x = -1 \).

2. **For \( 1 \leq x \leq 4 \), the function is \( f(x) = 1 \):**
   - This is a horizontal line at \( y = 1 \) from \( x = 1 \) to \( x = 4 \), including these endpoints.

3. **For \( x > 4 \), the function is \( f(x) = 0.5x - 1 \):**
   - This is a line with slope 0.5 and y-intercept -1.
   - The formula applies starting from \( x = 4 \), but \( x = 4 \) itself is not included.

### Step 2: Identify key points and intercepts

1. **x-intercept(s):**
   - For \( f(x) = x + 4 \) in the interval \( x < -1 \), set \( f(x) = 0 \):
     \[
     x + 4 = 0 \implies x = -4
     \]
     Thus, the x-intercept is at \( (-4, 0) \).
   - For \( f(x) = 0.5x - 1 \) in the interval \( x > 4 \), set \( f(x) = 0 \):
     \[
     0.5x - 1 = 0 \implies 0.5x = 1 \implies x = 2
     \]
     However, this x-intercept is not part of the defined domain for this piece of the function (because \( x > 4 \)).

2. **y-intercept:**
   - Set \( x = 0 \). Use the piece that applies for \( x = 0 \), which is \( f(x) = x + 4 \):
     \[
     f(0) = 0 + 4 = 4
     \]
     Thus, the y-intercept is \( (0, 4) \).

3. **Transition points:**
   - At \( x = -1 \), the function switches from \( f(x) = x + 4 \) to the next segment. Let's find the limit at this point:
     \[
     f(-1) = (-1) + 4 = 3
     \]
     So, \( f(x) \) approaches \( 3 \) from the left, but there is no function definition for \( x = -1 \), so it is an open point at \( (-1, 3) \).
   - At \( x = 1 \), the function switches to \( f(x) = 1 \). Therefore:
     \[
     f(1) = 1
     \]
     So, the point \( (1, 1) \) is on the graph.
   - At \( x = 4 \), the function switches to \( f(x) = 0.5x - 1 \). Let's compute the value:
     \[
     f(4) = 0.5(4) - 1 = 2 - 1 = 1
     \]
     So, the point \( (4, 1) \) is where the function transitions to the final piece.

### Step 3: Sketch and plot the points

1. Plot the line \( f(x) = x + 4 \) for \( x < -1 \), including the intercept at \( (-4, 0) \) and an open point at \( (-1, 3) \).
2. Draw the horizontal line \( f(x) = 1 \) from \( x = 1 \) to \( x = 4 \), including both endpoints.
3. Plot the line \( f(x) = 0.5x - 1 \) for \( x > 4 \), starting from \( (4, 1) \).

Would you like me to provide a visual sketch of the graph or further details?

### Follow-up questions:
1. How would you find the slope of each line segment in the piecewise function?
2. Can you verify continuity at \( x = 1 \) and \( x = 4 \)?
3. What is the domain and range of this function?
4. How would you describe the behavior of the function as \( x \to \infty \) and \( x \to -\infty \)?
5. What would happen if we shifted the piece \( 0.5x - 1 \) to start at a different point?

### Tip:
When sketching piecewise functions, always mark open and closed points clearly to show which values are included in the domain.

Sketch the graph of the piecewise function f(x), which is defined as:

f(x) =
{
  x + 4, when x < -1,
  1, when 1 ≤ x ≤ 4,
  0.5x - 1, when x > 4
}

The shape must be correct, and you must explicitly calculate and label the coordinates of any vertices, x-intercepts, and y-intercepts, as well as points where the formulas switch over.

Learn how to sketch the graph of a piecewise function with three parts, calculate x- and y-intercepts, and identify key transition points. This step-by-step guide covers the function f(x) = {x + 4, 1, 0.5x - 1}, including vertices and intercepts.

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