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The given function is a piecewise-defined function. Let's break it down:

\[
f(x) = \begin{cases} 
x - 4 & \text{if } x \leq 0 \\
\frac{4}{3}x - 2 & \text{if } x > 0 
\end{cases}
\]

### Steps to graph this function:

1. **For \( x \leq 0 \):** 
   - The function is \( f(x) = x - 4 \), which is a linear equation with slope 1 and y-intercept -4.
   - You can start plotting from \( x = 0 \), which gives \( f(0) = 0 - 4 = -4 \), and plot additional points going left.
   - The graph will have a closed dot at \( (0, -4) \) because of the "less than or equal to" condition.

2. **For \( x > 0 \):**
   - The function is \( f(x) = \frac{4}{3}x - 2 \), which is also a linear equation with slope \( \frac{4}{3} \) and y-intercept -2.
   - Start plotting from \( x = 0 \), but this time \( f(0) = -2 \). Since \( x = 0 \) is not included in this piece, this point will have an open dot.
   - Continue plotting points for values of \( x > 0 \).

### Plotting Points:
- For \( x \leq 0 \) (use points like \( x = -2 \), \( x = -1 \), etc.):
  - \( f(-2) = -2 - 4 = -6 \)
  - \( f(-1) = -1 - 4 = -5 \)
  
- For \( x > 0 \) (use points like \( x = 1 \), \( x = 2 \), etc.):
  - \( f(1) = \frac{4}{3}(1) - 2 = \frac{4}{3} - 2 = -\frac{2}{3} \)
  - \( f(3) = \frac{4}{3}(3) - 2 = 4 - 2 = 2 \)

### Graphing Instructions:
- Mark the closed point \( (0, -4) \) for \( x \leq 0 \).
- Mark the open point \( (0, -2) \) for \( x > 0 \).
- Draw the lines for each piece according to their respective rules.

Would you like further details or have any questions on this?

Here are five related questions:
1. How do you determine the slope of a piecewise linear function?
2. What is the difference between open and closed points in graphing?
3. How do you determine the domain of a piecewise function?
4. How would the graph change if the condition for \( x \leq 0 \) was instead \( x < 0 \)?
5. How do you graph functions with fractional slopes like \( \frac{4}{3}x \)?

**Tip:** When plotting piecewise functions, always pay attention to whether endpoints are open or closed, as this affects continuity.

Graph this function: f(x) = { x - 4, if x ≤ 0; 4/3x - 2, if x > 0 }

This problem involves graphing a piecewise function. The first part, f(x) = x - 4, applies for x ≤ 0, while the second part, f(x) = (4/3)x - 2, applies for x > 0. Learn how to graph both sections, including handling open and closed points at x = 0.

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