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We are given a piecewise function \( f(x) \) defined as:

\[
f(x) = \begin{cases}
-5x + 3 & \text{if } x \leq 2 \\
x - 4 & \text{if } x > 2
\end{cases}
\]

### Step 1: Graph the function
We need to plot the two segments of the piecewise function.

- **For \( x \leq 2 \)**, the equation is \( f(x) = -5x + 3 \).
  - The slope is -5, and the y-intercept is 3.
  - At \( x = 2 \), \( f(2) = -5(2) + 3 = -7 \). So, this point is \( (2, -7) \).

- **For \( x > 2 \)**, the equation is \( f(x) = x - 4 \).
  - The slope is 1, and the y-intercept is -4.
  - At \( x = 2 \), this part of the function is not defined, but at \( x = 2^+ \), the limit would approach \( 2 - 4 = -2 \).

### Step 2: Continuity Check
For a function to be continuous at \( x = 2 \), three conditions must be satisfied:
1. \( f(2) \) must exist.
2. The left-hand limit \( \lim_{x \to 2^-} f(x) \) must exist.
3. The right-hand limit \( \lim_{x \to 2^+} f(x) \) must exist, and both limits must be equal.

- \( f(2) = -7 \) (from the first piece).
- Left-hand limit \( \lim_{x \to 2^-} (-5x + 3) = -7 \).
- Right-hand limit \( \lim_{x \to 2^+} (x - 4) = -2 \).

Since the left-hand limit (-7) and the right-hand limit (-2) are not equal, the function is **not continuous at \( x = 2 \)**.

Would you like more details on any of the steps or graph plotting? 

Here are some related questions for further practice:
1. What are the conditions for a function to be continuous at a point?
2. How do you compute the limits from the left and right for a piecewise function?
3. Can you have continuity at a point where the function is not defined?
4. What is the graphical interpretation of a discontinuity in a piecewise function?
5. How can you modify the given piecewise function to make it continuous?

**Tip**: Always verify both left and right limits when checking for continuity at a point!

Suppose that the function f is defined for all real numbers as follows: f(x) = -5x + 3 if x ≤ 2 and f(x) = x - 4 if x > 2. Graph the function and determine whether or not it is continuous.

Learn how to check the continuity of the piecewise function f(x) = -5x + 3 if x ≤ 2 and f(x) = x - 4 if x > 2. Step-by-step graphing and limit calculations are used to determine if the function is continuous at x = 2. Suitable for high school students in Grades 10-12.

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