|x+3|/(x+3) for this piecewise, how do i know which one is inclusive -3 and thenone which is not inclusive

The expression \(\frac{|x+3|}{x+3}\) can be analyzed by determining the behavior of the absolute value function based on the value of \(x+3\). This will help us write the piecewise function and determine where to include or exclude specific points.

### 1. Behavior of \(\frac{|x+3|}{x+3}\):

- **When \(x + 3 > 0\) (i.e., \(x > -3\))**, \(|x + 3| = x + 3\). In this case, the expression simplifies as:
  \[
  \frac{|x+3|}{x+3} = \frac{x+3}{x+3} = 1
  \]

- **When \(x + 3 < 0\) (i.e., \(x < -3\))**, \(|x + 3| = -(x + 3)\). In this case, the expression becomes:
  \[
  \frac{|x+3|}{x+3} = \frac{-(x+3)}{x+3} = -1
  \]

- **When \(x = -3\)**, the denominator becomes zero, leading to an undefined value, since division by zero is not allowed.

### Piecewise Definition of \(\frac{|x+3|}{x+3}\):

\[
\frac{|x+3|}{x+3} = 
\begin{cases} 
1 & \text{if } x > -3 \\
-1 & \text{if } x < -3
\end{cases}
\]
And at \(x = -3\), the function is undefined, so \(-3\) is not included in either part of the piecewise function.

### Inclusion or Exclusion of \(-3\):

- For \(x > -3\), the value \(1\) is inclusive of all values greater than \(-3\), but not \(-3\) itself.
- For \(x < -3\), the value \(-1\) is inclusive of all values less than \(-3\), but \(-3\) itself is excluded because the function is undefined there.

So, \(-3\) is **not included** in either the positive or negative case.

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Would you like more details, or do you have any questions? Here are some related questions to consider:

1. What happens if we add a constant to the denominator?
2. How do absolute value functions behave in piecewise form for different ranges?
3. Can we extend the domain of the function to include \(x = -3\)?
4. What are some applications of absolute value functions like this one?
5. How does the graph of this piecewise function look?

**Tip:** Always check for points where the denominator equals zero when dealing with rational functions, as these are points of discontinuity or undefined behavior.

|x+3|/(x+3) for this piecewise, how do I know which one is inclusive of -3 and the one which is not inclusive?

Discover how the piecewise function |x+3|/(x+3) behaves around x = -3. Understand the behavior of the absolute value and rational functions, including discontinuity at x = -3, and learn which intervals include or exclude this point. Suitable for Grades 10-12.

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