Given the piecewise defined function:f(x)={x−2ifx≤2−2x+3ifx>2f(x)={x-2ifx≤2-2x+3ifx>2

Let's analyze the piecewise defined function \(f(x)\):

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

### Step-by-Step Analysis

1. **For \(x \leq 2\):**  
   The function is defined as \(f(x) = x - 2\).  
   This is a linear equation with a slope of 1 and a y-intercept of -2. It will be a straight line for all values of \(x\) less than or equal to 2.

2. **For \(x > 2\):**  
   The function is defined as \(f(x) = -2x + 3\).  
   This is also a linear equation but with a slope of -2 and a y-intercept of 3. It will be a straight line for all values of \(x\) greater than 2.

### Points of Interest

- At \(x = 2\), we need to check the value of both pieces to ensure continuity or identify a jump.

   - For \(x \leq 2\): \(f(2) = 2 - 2 = 0\)
   - For \(x > 2\): We check the value as we approach from the right side.
     \[
     \lim_{x \to 2^+} f(x) = -2(2) + 3 = -4 + 3 = -1
     \]

### Conclusion:

- **At \(x = 2\),** the function \(f(x) = x - 2\) gives \(f(2) = 0\).  
- As \(x\) approaches 2 from the right, \(f(x) = -2x + 3\) gives \(-1\).

Thus, there is a **jump discontinuity** at \(x = 2\) since the left-hand limit (\(0\)) and right-hand limit (\(-1\)) do not match.

### Graphical Interpretation:
- **For \(x \leq 2\):** Plot the line \(y = x - 2\) up to \(x = 2\) inclusive. This line passes through points \((0, -2)\) and \((2, 0)\).
- **For \(x > 2\):** Plot the line \(y = -2x + 3\) starting just after \(x = 2\). This line starts just after \((2, -1)\) and has a negative slope.

### Graph Behavior:
- The graph will have a continuous line segment from \(x = -\infty\) to \(x = 2\) (inclusive) following \(y = x - 2\).
- After \(x = 2\), the graph jumps to \((2, -1)\) and continues along the line \(y = -2x + 3\).

Would you like further details or have any questions?

Here are 5 related questions for further practice:
1. What is the range of \(f(x)\) given this piecewise function?
2. How would the graph change if the condition was \(x < 2\) instead of \(x \leq 2\)?
3. Determine the points of intersection for both linear parts of the piecewise function.
4. How do you determine if a piecewise function is continuous at a point?
5. What changes if the second piece is \(f(x) = -2x + 5\) instead of \(f(x) = -2x + 3\)?

**Tip:** Always check the endpoint and transition points in piecewise functions to determine continuity and to find exact function values.

Explore the piecewise defined function f(x) = x - 2 if x ≤ 2 and f(x) = -2x + 3 if x > 2 with detailed step-by-step analysis. Understand continuity and discontinuity at transition points. Suitable for advanced high school students.

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