The function $f(x) = |x + 1|$ represents an absolute value function, which behaves differently based on whether the expression inside the absolute value, $x + 1$, is positive or negative.

Analysis of $f(x) = |x + 1|$:

1. Piecewise Definition: We can break down the absolute value function into a piecewise function:

   \begin{cases} x + 1 & \text{if } x + 1 \geq 0 \\ -(x + 1) & \text{if } x + 1 < 0 \end{cases}$$ Simplifying the conditions, we get: $$f(x) = \begin{cases} x + 1 & \text{if } x \geq -1 \\ -x - 1 & \text{if } x < -1 \end{cases}$$

2. Graph Shape: The graph of $f(x) = |x + 1|$ is a V-shaped graph, with its vertex (point of intersection) at $x = -1$, where $f(-1) = 0$.

3. Domain and Range:

   • Domain: The domain of $f(x)$ is all real numbers, $x \in (-\infty, \infty)$, since absolute value functions are defined everywhere.
   • Range: The range of $f(x)$ is $[0, \infty)$ because absolute values are always non-negative, and the minimum value $f(x)$ can attain is 0.

4. Continuity and Differentiability:

   • Continuity: $f(x)$ is continuous across its domain.
   • Differentiability: $f(x)$ is not differentiable at $x = -1$, as there is a sharp corner (non-smooth transition) at this point.

Would you like more details, or do you have any specific questions?

Related Questions:

1. How does shifting an absolute value function horizontally affect its graph?
2. What is the derivative of $f(x) = |x + 1|$ where it is differentiable?
3. How would the graph change if the function were $f(x) = |x - 1|$?
4. Can you find the inverse of $f(x) = |x + 1|$ if it exists?
5. How does the graph of $f(x) = |x + 1|$ relate to $f(x) = |x|$?

Tip:

Absolute value functions are defined piecewise, making them excellent examples for understanding continuity and differentiability concepts.