Sketch the graph of a function f that satisfies all of the given conditions.lim x→6+ f(x) = 7, lim x→6− f(x) = 5, lim x→−2 f(x) = 5, f(6) = 6, f(−2) = 4

To sketch the graph of the function $f(x)$ that satisfies the provided conditions, we can break down the requirements and analyze each one.

1. Break Down the Problem

1. Limits at $x = 6$:

   • $\lim_{x \to 6^+} f(x) = 7$ (right-hand limit)
   • $\lim_{x \to 6^-} f(x) = 5$ (left-hand limit)
   • $f(6) = 6$ (the value of the function at $x = 6$)

2. Limit at $x = -2$:

   • $\lim_{x \to -2} f(x) = 5$ (two-sided limit)
   • $f(-2) = 4$ (the value of the function at $x = -2$)

2. Relevant Concepts

• To represent the limits and function values, we will use:
  • Open circles to denote limits where the function does not equal the limit.
  • Solid dots to indicate the actual value of the function at certain points.

3. Analysis and Detail

1. At $x = 6$:

   • As $x$ approaches $6$ from the right, $f(x)$ approaches $7$, but at $x = 6$ itself, the function takes the value $6$.
   • Therefore, we will place an open circle at the point $(6, 7)$ and a solid dot at $(6, 6)$.

2. At $x = -2$:

   • As $x$ approaches $-2$, $f(x)$ approaches $5$, but at $x = -2$, the value of the function is $4$.
   • We will place an open circle at the point $(-2, 5)$ and a solid dot at $(-2, 4)$.

3. Limit at $-2$ also sets the behavior of the graph approaching this point from both the left and the right.

4. Verify and Summarize

• We have established the limits and values:
  • $(6, 7)$ is an open circle, $(6, 6)$ is a solid dot.
  • $(-2, 5)$ is an open circle, $(-2, 4)$ is a solid dot.
• The function should be continuous except at the specified points.

Final Answer

The sketch of the function $f(x)$ will include:

• An open circle at $(6, 7)$.
• A solid dot at $(6, 6)$.
• An open circle at $(-2, 5)$.
• A solid dot at $(-2, 4)$.
• The graph can be further drawn such that it approaches $5$ as $x$ approaches $-2$ from both sides, and shows the different behaviors around $x = 6$.

Please visualize the graph based on the description above, considering how the function behaves at and around those key points.