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 $\frac{|x+3|}{x+3}$:

\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$$ is **not included** in either the positive or negative case. --- Would you like more details, or do you have any questions? Here are some related questions to consider: 1. What happens if we add a constant to the denominator? 2. How do absolute value functions behave in piecewise form for different ranges? 3. Can we extend the domain of the function to include $$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.