It seems you’ve uploaded an image containing some mathematical problems. Let me analyze the content of the image and provide solutions or explanations. One moment while I process it.The image contains a piecewise function $f(x)$ and conditions involving $x$ and the function's outputs. I'll break it down into the key parts I see and provide explanations for them.

Key Observations:

1. Piecewise Function:
   • The function $f(x)$ appears to be defined differently for $x \leq 3$ and $x > 3$. Specifically: \begin{cases} x - 3, & x \leq 3 \\ \frac{1}{x - 3}, & x > 3 \end{cases}$$
2. Two tasks seem to be highlighted:
   • One task involves determining a parameter $a$, where $a$ might correspond to certain conditions on the piecewise function or limits of $x$.
   • The answers for $a$ are marked in the choices $A = -4, B = 4, C = -\frac{1}{4}, D = \frac{1}{4}$.
   • Another section involves determining the correct intervals or outputs for specific $x$ values.

Analysis of the Problem:

Part 1: Behavior of $f(x)$

• For $x \leq 3$: The function simplifies to $f(x) = x - 3$. This is a linear function with a slope of 1 and a y-intercept at -3.
• For $x > 3$: The function is $f(x) = \frac{1}{x - 3}$, a hyperbolic function that becomes undefined as $x \to 3^+$ (from the right of 3).

Part 2: Interpreting $a$

The problem likely involves ensuring continuity, limits, or certain evaluations of $f(x)$ at a specific value of $x$ or over an interval. Let me know if you want me to compute any specific property like limits or derivatives, or if you’d like further clarification about $f(x)$.

Would you like detailed steps for solving any specific part, or should I analyze $a$ in depth?

Follow-Up Questions:

1. Would you like to explore the limit of $f(x)$ as $x \to 3^+$ and $x \to 3^-$?
2. Do you want me to calculate whether $f(x)$ is continuous at $x = 3$?
3. Shall I help evaluate specific $x$-values for $f(x)$?
4. Would you like clarification on how the hyperbolic function $\frac{1}{x-3}$ behaves for $x > 3$?
5. Do you want to confirm why the piecewise definitions split at $x = 3$?

Tip:

Always check continuity and differentiability when working with piecewise functions. It helps identify whether any "jumps" or undefined behaviors occur at the split points.