The problem asks for a sketch of a function $y = f(x)$ based on a set of given conditions:

1. Continuity and Discontinuity:

   • $f(x)$ is continuous on $[-2, 0) \cup (0, 1) \cup (1, \infty)$, meaning there are no breaks in the graph on these intervals.
   • $f(x)$ is discontinuous at $x = 0$ and $x = 1$, implying breaks or jumps at these points.

2. Asymptotes:

   • Horizontal asymptote at $y = 3$, meaning the graph approaches $y = 3$ as $x \to \infty$.
   • Vertical asymptote at $x = 0$, meaning the function grows unbounded near $x = 0$.

3. Limits and Function Values:

   • $\lim_{x \to 1^-} f(x) = 1$, implying the left-hand limit as $x$ approaches 1 is 1.
   • $\lim_{x \to 1^+} f(x) = 2$, implying the right-hand limit as $x$ approaches 1 is 2.
   • $f(1) = 3$, implying a point on the graph at $x = 1$ and $f(x) = 3$.

Interpretation:

• For $x = 0$: The function has a vertical asymptote, which means the function may shoot up to infinity or down to negative infinity as it nears $x = 0$, but the function doesn't exist at $x = 0$.

• For $x = 1$: The function is discontinuous here. The left-hand limit approaches 1, the right-hand limit approaches 2, and the actual value of the function at $x = 1$ is 3. This suggests a jump discontinuity at $x = 1$, with a value marked at 3.

• Horizontal asymptote at $y = 3$: As $x \to \infty$, the function approaches $y = 3$, meaning the graph levels off at this value but never touches it.

Sketch Approach:

• Start by drawing a vertical asymptote at $x = 0$.
• On the left of $x = 0$, sketch a function continuous from $x = -2$ to 0, approaching the vertical asymptote.
• After $x = 0$, sketch the function so that it approaches the vertical asymptote but remains continuous until $x = 1$.
• At $x = 1$, there is a jump discontinuity: the graph jumps from 1 on the left to 2 on the right and marks a point at $(1, 3)$.
• Finally, for $x > 1$, the graph should approach the horizontal asymptote at $y = 3$ as $x \to \infty$.

Would you like more details on how to draw the graph clearly?

Additional Questions:

1. What is the significance of horizontal asymptotes in function behavior?
2. How would the graph behave as $x \to -\infty$?
3. How does the discontinuity at $x = 1$ affect the graph's shape?
4. Could the vertical asymptote behavior differ depending on the direction $x \to 0$?
5. How would you plot the function between $-2$ and $0$ on this graph?

Tip: Asymptotes indicate the direction a graph heads but doesn't cross or touch at infinity (horizontal) or a specific $x$-value (vertical).