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The piecewise function given is:

\[
f(x) = 
\begin{cases} 
x - 4 & \text{if } x < 2, \\
-2x + 2 & \text{if } x \geq 2. 
\end{cases}
\]

### 1. Graph of the Function:
To graph the function, let's consider each piece separately:

1. **For \( x < 2 \):** The function is \( f(x) = x - 4 \). This is a linear function with a slope of 1 and y-intercept at \(-4\). 

2. **For \( x \geq 2 \):** The function is \( f(x) = -2x + 2 \). This is also a linear function with a slope of \(-2\) and y-intercept at 2. 

### 2. Continuity Check:
To determine whether the function is continuous at \( x = 2 \), we need to check:

- **Limit from the left (\( x \to 2^- \))**: 
  \[
  \lim_{x \to 2^-} f(x) = 2 - 4 = -2.
  \]

- **Limit from the right (\( x \to 2^+ \))**:
  \[
  \lim_{x \to 2^+} f(x) = -2(2) + 2 = -2.
  \]

- **Value of the function at \( x = 2 \)**:
  \[
  f(2) = -2(2) + 2 = -2.
  \]

Since the left-hand limit, right-hand limit, and the value of the function at \( x = 2 \) are all equal, \( f(x) \) is continuous at \( x = 2 \). Thus, the function \( f(x) \) is continuous everywhere.

Would you like me to graph the function or provide more details? 

### Related Questions:
1. What is the slope of each piece of the function?
2. How do you determine the continuity of a piecewise function?
3. How would the graph change if one of the inequalities were strict?
4. Can a function be discontinuous even if the left-hand and right-hand limits are equal?
5. What are the domain and range of this piecewise function?

### Tip:
For piecewise functions, always check the function's value and limits at the boundary points to determine continuity.

Suppose that the function f is defined, for all real numbers, as follows: f(x) = { x - 4 if x < 2, -2x + 2 if x ≥ 2. Graph the function f. Then determine whether or not the function is continuous.

This problem involves graphing a piecewise function and determining its continuity. Explore limits from the left and right, calculate the function value at x = 2, and confirm if the function is continuous. Suitable for Grades 10-12 students studying algebra and calculus concepts.

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