https://file.aiquickdraw.com/mathbot/file/1726895014027c51cbjuh.jpg

Let's discuss the continuity of the given function:

\[
f(x) = \frac{x + 2}{x - 3}
\]

### 1. Continuity on the open interval \((-2 < x < 3)\)

- The function \(f(x)\) is a rational function, which is continuous everywhere in its domain (where the denominator is not zero).
- The function has a discontinuity where the denominator \(x - 3 = 0\), i.e., at \(x = 3\), because dividing by zero is undefined.
- On the interval \((-2, 3)\), the denominator does not become zero, so the function is continuous everywhere on this open interval.

### 2. Continuity on the closed interval \([-2 \leq x \leq 3]\)

- The closed interval includes the point \(x = 3\), which is a point of discontinuity for the function.
- Therefore, \(f(x)\) is not continuous on the closed interval \([-2, 3]\) because of the discontinuity at \(x = 3\).

### Conclusion:
- **On the open interval \((-2 < x < 3)\)**: The function is continuous.
- **On the closed interval \([-2 \leq x \leq 3]\)**: The function is not continuous because it has a discontinuity at \(x = 3\).

Would you like further details on this or have any other questions?

Here are 5 related questions for deeper exploration:
1. What is a removable discontinuity, and does \(x = 3\) represent one?
2. How can we find the limit of \(f(x)\) as \(x\) approaches 3?
3. What happens to the value of \(f(x)\) as \(x\) approaches 3 from the left and right?
4. What does it mean for a function to be continuous at a point?
5. How would you graph this function, showing its points of discontinuity?

**Tip:** When checking for continuity, always evaluate the domain of the function first, and then identify where it might fail to be continuous.

Discuss the continuity of the function f(x) = (x + 2) / (x - 3) on the open interval (-2 < x < 3) and the closed interval [-2 ≤ x ≤ 3].

Explore the continuity of the rational function f(x) = (x + 2) / (x - 3) on both open and closed intervals. Understand where the function is continuous and where it has a discontinuity. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation