Find each of the following functions and state their domains. (Enter the domains in interval notation.)
f(x) = 

3 − x
,    g(x) = 

x2 − 4


Let's analyze each function step by step.

### 1. \( f(x) = 3 - x \)

- **Domain of \( f(x) \):**
    - Since \( f(x) = 3 - x \) is a linear function, there are no restrictions on the value of \( x \) (i.e., no denominators, square roots, or logarithms that could create undefined points).
    - The domain is all real numbers \( x \in (-\infty, \infty) \).

- **Domain of \( f(x) \) in interval notation:**
    - \( (-\infty, \infty) \)

### 2. \( g(x) = \frac{1}{x^2 - 4} \)

- **Domain of \( g(x) \):**
    - This is a rational function. For rational functions, the denominator must not be zero.
    - So, find when the denominator is zero:
      \[
      x^2 - 4 = 0
      \]
      Solving this:
      \[
      x^2 = 4 \implies x = \pm 2
      \]
      Therefore, \( g(x) \) is undefined at \( x = 2 \) and \( x = -2 \).

- **Domain of \( g(x) \) in interval notation:**
    - The domain is all real numbers except \( x = 2 \) and \( x = -2 \), so the domain in interval notation is:
    \[
    (-\infty, -2) \cup (-2, 2) \cup (2, \infty)
    \]

### Final Answer:

- Domain of \( f(x) = 3 - x \): \( (-\infty, \infty) \)
- Domain of \( g(x) = \frac{1}{x^2 - 4} \): \( (-\infty, -2) \cup (-2, 2) \cup (2, \infty) \)

Would you like further details or clarifications on these functions?

---

Here are five related questions:
1. What is the range of \( f(x) = 3 - x \)?
2. How would the domain change if \( f(x) \) had a square root?
3. Can you explain how vertical asymptotes relate to the domain of \( g(x) \)?
4. What happens to the domain of \( g(x) \) if we change the denominator to \( x^2 - 9 \)?
5. How does adding a constant to \( g(x) \) affect its domain?

**Tip**: Always check the denominator of a rational function for zero values to avoid undefined points!

Find each of the following functions and state their domains. (Enter the domains in interval notation.) f(x) = 3 − x, g(x) = x^2 − 4

Understand how to find the domains of the functions f(x) = 3 - x and g(x) = 1 / (x^2 - 4). This problem covers essential algebraic concepts like linear functions and rational functions. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation