https://r2.erweima.ai/mathbot/image/1724931862190dpug33po.jpg

To solve part (1) of the question from the image, let's go through the steps.

### Problem Recap:
You are asked to determine the domain of the function \( f(x) \), which is given as:
\[
f(x) = \frac{-b^2}{4 - x^2} + 3
\]
where \( b \) is a constant.

### Solution for (1): Determining the Domain of \( f(x) \)

The domain of a function is the set of all possible values of \( x \) for which the function is defined.

1. **Identify Potential Issues:**
   - The function \( f(x) \) involves a fraction. For the function to be defined, the denominator \( 4 - x^2 \) must not be zero.

2. **Set the Denominator Not Equal to Zero:**
   \[
   4 - x^2 \neq 0
   \]
   Solve for \( x \):
   \[
   x^2 \neq 4
   \]
   Taking the square root of both sides:
   \[
   x \neq \pm 2
   \]

3. **Conclusion:**
   The function \( f(x) \) is undefined at \( x = 2 \) and \( x = -2 \) because these values make the denominator zero. Therefore, the domain of \( f(x) \) is all real numbers except \( x = \pm 2 \).

   **Domain:**
   \[
   x \in \mathbb{R}, \quad x \neq \pm 2
   \]

### Summary:
The domain of \( f(x) \) is all real numbers except \( x = 2 \) and \( x = -2 \).

---

Would you like a detailed explanation of this solution or any additional assistance? Here are some follow-up questions you might find useful:

1. How do you find the range of the function \( f(x) \)?
2. What happens to the function \( f(x) \) near \( x = \pm 2 \)?
3. Can you graph \( f(x) \) to visualize the points where the function is undefined?
4. How would the domain change if \( b \) were a specific value like 0?
5. How do different values of \( b \) affect the graph of \( f(x) \)?

**Tip:** When determining the domain of a function, always check for values that might cause division by zero or other undefined operations like square roots of negative numbers.

Learn how to determine the domain of the function f(x) = -b^2 / (4 - x^2) + 3. This problem involves analyzing a fractional function and identifying the set of all possible values of x for which the function is defined. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation