what is the y equation of f(x)=abs(x+3)-2

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:
\[
f(x) = 
\begin{cases} 
x + 1 & \text{if } x \geq -3, \\
-x - 5 & \text{if } x < -3.
\end{cases}
\]

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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.

Explore the absolute value function f(x) = |x + 3| - 2 in detail. Learn how to express it as a piecewise function and understand its behavior. Suitable for students in Grades 9-12.

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