The function given is $f(x) = |x + 3| - 2$. To express this function without the absolute value, we consider two cases based on the expression inside the absolute value:

Case 1: $x + 3 \geq 0$ (i.e., $x \geq -3$)

In this case, $|x + 3| = x + 3$, so the function becomes: $f(x) = (x + 3) - 2 = x + 1$ Thus, for $x \geq -3$, $f(x) = x + 1$.

Case 2: $x + 3 < 0$ (i.e., $x < -3$)

In this case, $|x + 3| = -(x + 3) = -x - 3$, so the function becomes: $f(x) = (-x - 3) - 2 = -x - 5$ Thus, for $x < -3$, $f(x) = -x - 5$.

Summary

The piecewise function is:

\begin{cases} x + 1 & \text{if } x \geq -3, \\ -x - 5 & \text{if } x < -3. \end{cases}$$ Would you like further details or clarification on this? Here are 5 related questions you might find interesting: 1. How do you graph a piecewise function like this one? 2. What is the vertex of the function $$f(x) = |x + 3| - 2$$ and how is it determined? 3. How do you determine the range of $$f(x) = |x + 3| - 2$$? 4. What happens to the graph of $$|x|$$ when you add or subtract a constant inside the absolute value? 5. How do transformations affect the graph of a function like $$f(x) = |x|$$? **Tip:** When dealing with absolute value functions, always consider breaking down the problem into cases where the expression inside the absolute value is positive or negative.