The absolute value function $f(x) = |x + 1|$ can be written as a piecewise function. The absolute value function will change its behavior depending on whether the expression inside the absolute value is positive or negative.

Here's how you can write it as a piecewise function:

\begin{cases} x + 1 & \text{if } x + 1 \geq 0 \\ -(x + 1) & \text{if } x + 1 < 0 \end{cases}$$ This can be simplified further by solving the inequalities: $$f(x) = \begin{cases} x + 1 & \text{if } x \geq -1 \\ -(x + 1) & \text{if } x < -1 \end{cases}$$ ### Summary: - When $$x \geq -1$$, $$f(x) = x + 1$$. - When $$x < -1$$, $$f(x) = -(x + 1) = -x - 1$$. Would you like further details or have any questions? Here are some related questions: 1. How do you graph a piecewise function? 2. What are the key points on the graph of $$f(x) = |x + 1|$$? 3. How can we determine where the function $$f(x)$$ is continuous? 4. What happens if you shift the absolute value function horizontally? 5. Can you solve for $$x$$ when $$f(x) = 3$$ for this function? **Tip:** When dealing with absolute value functions, it's helpful to first identify the point where the expression inside the absolute value equals zero. This is where the function typically changes its behavior.