To solve the given problems using the graph of the function $f$, let's analyze each part one by one:

Given Parts:

1. $f(1)$
2. $\lim_{{x \to 1}} f(x)$
3. $f(6)$
4. $\lim_{{x \to 6}} f(x)$

Analyzing the Graph:

Let's examine the graph:

1. Point (a): $f(1)$

   • To find $f(1)$, we look for the y-coordinate of the function at $x = 1$.
   • From the graph, there is an open circle at $x = 1$ with $y = 3$, meaning $f(1)$ is undefined at that point.
   • Answer: DNE (Does Not Exist).

2. Point (b): $\lim_{{x \to 1}} f(x)$

   • To find the limit as $x \to 1$, we check the values of the function as $x$ approaches 1 from both the left and right.
   • As $x \to 1$ from both sides, the function approaches $y = 3$.
   • Answer: 3.

3. Point (c): $f(6)$

   • To find $f(6)$, we look for the y-coordinate of the function at $x = 6$.
   • The graph shows a point at $x = 6$ with $y = 4$ (a filled circle).
   • Answer: 4.

4. Point (d): $\lim_{{x \to 6}} f(x)$

   • To find the limit as $x \to 6$, we check the values of the function as $x$ approaches 6 from both sides.
   • As $x \to 6$, the function approaches $y = 2$ from both sides (based on the continuous line before reaching $x = 6$).
   • Answer: 2.

Final Answers:

(a) $f(1) = \text{DNE}$
(b) $\lim_{{x \to 1}} f(x) = 3$
(c) $f(6) = 4$
(d) $\lim_{{x \to 6}} f(x) = 2$

Would you like more details on any part of the solution, or have any further questions?

Related Questions:

1. What is the difference between a limit and the value of a function at a point?
2. How can discontinuities in a function be classified?
3. What is a removable discontinuity, and how does it appear on a graph?
4. What are the left-hand limit and right-hand limit, and how do they relate to the overall limit?
5. How can you determine if a function is continuous at a certain point?

Tip: When analyzing limits from a graph, always check for open and closed circles to understand where the function value is defined or undefined.