Find each of the following functions and state their domains. (Enter the domains in interval notation.)
f(x) = (3 − x)^1/2,    g(x) = (x^2 − 4)^1/2
f + g = ?         domain = ?


Let's start by analyzing both functions and finding their domains.

### 1. **For \( f(x) = \sqrt{3 - x} \):**
- The square root function \( \sqrt{x} \) is only defined when the argument is non-negative. 
- Therefore, we need \( 3 - x \geq 0 \), which simplifies to:
  \[
  x \leq 3
  \]
- Hence, the domain of \( f(x) \) is all \( x \) such that \( x \leq 3 \), which in interval notation is:
  \[
  \text{Domain of } f(x): (-\infty, 3]
  \]

### 2. **For \( g(x) = \sqrt{x^2 - 4} \):**
- Again, the square root function is only defined when the argument is non-negative.
- We need \( x^2 - 4 \geq 0 \), which simplifies to:
  \[
  x^2 \geq 4
  \]
  This inequality holds when \( x \leq -2 \) or \( x \geq 2 \). 
- So, the domain of \( g(x) \) is \( x \in (-\infty, -2] \cup [2, \infty) \).

### 3. **For \( (f + g)(x) \):**
- The function \( (f + g)(x) \) will be defined wherever both \( f(x) \) and \( g(x) \) are defined.
- Therefore, the domain of \( f + g \) is the intersection of the domains of \( f(x) \) and \( g(x) \).

#### Intersection of Domains:
- Domain of \( f(x) \): \( (-\infty, 3] \)
- Domain of \( g(x) \): \( (-\infty, -2] \cup [2, \infty) \)

The intersection of these two domains is:
- For \( (-\infty, 3] \) and \( (-\infty, -2] \), the overlap is \( (-\infty, -2] \).
- For \( (-\infty, 3] \) and \( [2, \infty) \), the overlap is \( [2, 3] \).

Thus, the domain of \( f + g \) is:
\[
(-\infty, -2] \cup [2, 3]
\]

### 4. **Conclusion:**
- \( (f + g)(x) = \sqrt{3 - x} + \sqrt{x^2 - 4} \)
- Domain of \( f + g \): \( (-\infty, -2] \cup [2, 3] \)

Would you like further details on this? Here are some related questions you might find useful:

1. What is the graph of \( f(x) = \sqrt{3 - x} \)?
2. How do you solve inequalities involving square roots?
3. Can the domain of \( f(x) \) and \( g(x) \) overlap at specific points?
4. What happens when the square root is complex (i.e., negative inside)?
5. How do you compute the sum of two functions in general?

**Tip:** Always check for restrictions such as non-negativity for functions like square roots and logarithms.

Find each of the following functions and state their domains. (Enter the domains in interval notation.) f(x) = (3 − x)^1/2, g(x) = (x^2 − 4)^1/2 f + g = ? domain = ?

Learn how to find the domains of f(x) = √(3 - x) and g(x) = √(x^2 - 4), and compute the sum of the functions f + g. This problem involves understanding domain restrictions for square root functions and finding the intersection of domains. Suitable for students in Grades 10-12.

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