To solve this problem, let's analyze the graph and compare it with the given options.

Observation:

The graph shows:

1. A straight line with slope $+1$ (upward) for $x > 0$.
2. A straight line with slope $-1$ (downward) for $x < 0$.

This corresponds to a function where the absolute value affects the slope:

• For $x \geq 0$, $y = x$.
• For $x < 0$, $y = -x$.

Match with the Options:

• Option A: $y = |x|$. Correct globally, but not restricted.
• Option B: $y = -x$. Only correct for $x < 0$, not globally correct.
• Option C: $y = |x|$ with $x \leq 0$. Incorrect, as this condition is contradictory to the behavior shown.
• Option D: $y = -x$ for $x < 0$. Correct conditionally, and matches the restriction shown.

Correct Answer:

D. $y = -x$ for $x < 0$.

Let me know if you'd like further explanation!

Here are related questions to expand your understanding:

1. What is the derivative of $y = |x|$ and how does it behave at $x = 0$?
2. How does restricting the domain of a function like $y = |x|$ affect its graph?
3. Can a piecewise function like $y = |x|$ be continuous and differentiable? Why?
4. What happens if we shift the graph of $y = |x|$ vertically or horizontally?
5. How can you rewrite $y = |x|$ using a piecewise function?

Tip: For any absolute value function, think of it as having two linear "pieces," one for $x \geq 0$ and one for $x < 0$.