Use the given functions f and g to find 
f + g,
 
f − g,
 
fg,
 and 
f
g
.
 State the domain of each. (Enter your answer for the domain in interval notation.)
f(x) = 

sqrt25 − x2
,    g(x) = 5 + x

We are given the functions:

- \( f(x) = \sqrt{25 - x^2} \)
- \( g(x) = 5 + x \)

We need to find:

1. \( (f + g)(x) = f(x) + g(x) \)
2. \( (f - g)(x) = f(x) - g(x) \)
3. \( (f \cdot g)(x) = f(x) \cdot g(x) \)
4. \( \left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} \)

We will also find the domain of each function and operation. Let's go through each step.

### 1. \( f + g \)

\[
(f + g)(x) = f(x) + g(x) = \sqrt{25 - x^2} + (5 + x)
\]

#### Domain of \( f + g \)
- For \( f(x) = \sqrt{25 - x^2} \), we need \( 25 - x^2 \geq 0 \), meaning \( x^2 \leq 25 \), so \( -5 \leq x \leq 5 \).
- \( g(x) = 5 + x \) has no restrictions, its domain is all real numbers.

Thus, the domain of \( f + g \) is \( [-5, 5] \).

### 2. \( f - g \)

\[
(f - g)(x) = f(x) - g(x) = \sqrt{25 - x^2} - (5 + x)
\]

#### Domain of \( f - g \)
- The domain of \( f - g \) is the same as the domain of \( f(x) \), which is \( [-5, 5] \).

### 3. \( fg \)

\[
(fg)(x) = f(x) \cdot g(x) = \sqrt{25 - x^2} \cdot (5 + x)
\]

#### Domain of \( fg \)
- The domain is again constrained by \( f(x) \), so the domain of \( fg \) is \( [-5, 5] \).

### 4. \( \frac{f}{g} \)

\[
\left( \frac{f}{g} \right)(x) = \frac{\sqrt{25 - x^2}}{5 + x}
\]

#### Domain of \( \frac{f}{g} \)
- In addition to \( -5 \leq x \leq 5 \), we need to exclude points where \( g(x) = 0 \), which happens when \( 5 + x = 0 \), or \( x = -5 \).

Thus, the domain of \( \frac{f}{g} \) is \( (-5, 5] \).

### Summary of Domains:
1. \( f + g \): Domain = \( [-5, 5] \)
2. \( f - g \): Domain = \( [-5, 5] \)
3. \( fg \): Domain = \( [-5, 5] \)
4. \( \frac{f}{g} \): Domain = \( (-5, 5] \)

Would you like more details on any step, or have any specific questions?

### Related Questions:
1. What are the restrictions on \( f(x) = \sqrt{25 - x^2} \) and why?
2. How do you find the domain of a square root function in general?
3. How does the domain of a quotient function differ from other operations?
4. Can you solve for specific values of \( f + g \) at \( x = 0 \) or other points?
5. What happens to the domain of \( f(x) \) when its form changes to a cubic root instead of a square root?

### Tip:
Always remember that the square root function requires the radicand to be non-negative, which affects the domain significantly.

Use the given functions f and g to find f + g, f − g, fg, and f/g. State the domain of each. f(x) = sqrt(25 − x^2), g(x) = 5 + x

Solve the function operations f + g, f − g, fg, and f/g using f(x) = sqrt(25 − x^2) and g(x) = 5 + x. Learn how to determine the domain for each function operation, covering key algebraic concepts and theorems. Suitable for Grades 10-12.

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