Let's break down the two problems one by one and outline how to sketch the graphs for both functions $f$ and $g$ based on the given conditions.

Problem 4: Sketch a graph of a function $f$ that satisfies the following conditions:

• Condition a: $f(-2) = 5$

  • The function passes through the point $(-2, 5)$.

• Condition b: $\lim_{x \to -2} f(x) = 1$

  • As $x$ approaches $-2$, the function approaches a value of 1. There could be a discontinuity at $x = -2$, implying the function jumps from $5$ to $1$.

• Condition c: $\lim_{x \to 4^-} f(x) = 3$

  • As $x$ approaches $4$ from the left, the function approaches a value of 3.

• Condition d: $f$ is increasing on $x < -2$

  • The function increases for all values less than $-2$.

• Condition e: $\lim_{x \to 4^-} f(x) < \lim_{x \to 4^+} f(x)$

  • This indicates a jump at $x = 4$, where the value to the left of $4$ (which is $3$) is less than the value to the right of $4$. So, the function jumps upward at $x = 4$.

Problem 5: Sketch a graph of a function $g$ that satisfies the following conditions:

• Condition a: $g(1) = 3$

  • The function passes through the point $(1, 3)$.

• Condition b: $\lim_{x \to 1} g(x) = -2$

  • As $x$ approaches $1$, the function approaches $-2$, implying there could be a jump discontinuity.

• Condition c: $\lim_{x \to -3^+} g(x) = 5$

  • As $x$ approaches $-3$ from the right, the function approaches a value of 5.

• Condition d: $g$ is increasing only on $-5 < x < -3$ and $x > 1$

  • The function is increasing between $-5$ and $-3$, and again for $x > 1$.

• Condition e: $\lim_{x \to -3^-} g(x) > \lim_{x \to -3^+} g(x)$

  • This indicates a jump discontinuity at $x = -3$, where the limit from the left is greater than the limit from the right, so the function jumps downward at $x = -3$.

Graph Interpretation:

For both problems, sketching the graphs would involve:

1. Marking important points: For example, in Problem 4, $(-2, 5)$ for $f$ and in Problem 5, $(1, 3)$ for $g$.
2. Handling limits: Adding jumps or discontinuities where limits don’t match the function’s value or where left and right limits differ (as in $x = -2$ for Problem 4 and $x = -3$ for Problem 5).
3. Indicating monotonicity: Showing the increasing behavior where indicated in the conditions.

Would you like me to plot these graphs for better clarity?

Relative Questions:

1. What is a jump discontinuity in a function, and how does it affect the graph?
2. How do limits from the left and right sides influence the shape of a function at a given point?
3. What are the conditions for a function to be increasing or decreasing over an interval?
4. How can we differentiate between continuous and discontinuous functions?
5. What role does the behavior of a function around a point (like limits) play in calculus?

Tip: When sketching functions with discontinuities, it's important to differentiate between the function's value at a point and the behavior as you approach that point from either side.