https://file.aiquickdraw.com/mathbot/file/17260120686475xjquti9.jpg

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.

Sketch a graph of a function f that satisfies the following conditions, and sketch a graph of a function g that satisfies another set of conditions (see attached image).

This problem involves sketching graphs for functions based on specific conditions, including limits, discontinuities, and increasing behavior. Ideal for students in Grades 11-12 or early college, it covers key calculus concepts such as limits and jump discontinuities.

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