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We are given the following functions and asked to find:

- \( f + g \)
- \( f - g \)
- \( fg \)
- \( \frac{f}{g} \)

The functions are:
- \( f(x) = \sqrt{x} \)
- \( g(x) = x - 17 \)

We'll also determine the domain of each resulting function.

### 1. \( f + g \)
\[
(f + g)(x) = f(x) + g(x) = \sqrt{x} + (x - 17)
\]
**Domain**: 
- \( f(x) = \sqrt{x} \) requires \( x \geq 0 \) because we cannot take the square root of a negative number.
- \( g(x) = x - 17 \) is defined for all real numbers.

Thus, the domain of \( f + g \) is \( [0, \infty) \).

### 2. \( f - g \)
\[
(f - g)(x) = f(x) - g(x) = \sqrt{x} - (x - 17) = \sqrt{x} - x + 17
\]
**Domain**:
- Same as \( f(x) \), \( x \geq 0 \).

Thus, the domain of \( f - g \) is \( [0, \infty) \).

### 3. \( fg \)
\[
(fg)(x) = f(x) \cdot g(x) = \sqrt{x} \cdot (x - 17) = \sqrt{x}(x - 17)
\]
**Domain**:
- Again, the square root function \( \sqrt{x} \) requires \( x \geq 0 \).

Thus, the domain of \( fg \) is \( [0, \infty) \).

### 4. \( \frac{f}{g} \)
\[
\left(\frac{f}{g}\right)(x) = \frac{\sqrt{x}}{x - 17}
\]
**Domain**:
- \( f(x) = \sqrt{x} \) requires \( x \geq 0 \).
- The denominator \( g(x) = x - 17 \neq 0 \), so \( x \neq 17 \).

Thus, the domain of \( \frac{f}{g} \) is \( [0, \infty) \setminus \{17\} \), meaning \( x \geq 0 \) but \( x \neq 17 \).

### Summary:
- \( f + g = \sqrt{x} + x - 17 \), domain: \( [0, \infty) \)
- \( f - g = \sqrt{x} - x + 17 \), domain: \( [0, \infty) \)
- \( fg = \sqrt{x}(x - 17) \), domain: \( [0, \infty) \)
- \( \frac{f}{g} = \frac{\sqrt{x}}{x - 17} \), domain: \( [0, \infty) \setminus \{17\} \)

Would you like more details or have any further questions?  
Here are some related questions:

1. What is the difference between the domain and range of a function?
2. How can you determine if a function is continuous or discontinuous?
3. How do we handle square root functions when combined with other operations?
4. Can you explain how division affects the domain of a function?
5. How would you graph these combined functions to visualize their domains?

**Tip**: When dealing with square roots, always ensure the expression under the root is non-negative to determine the function's domain.

Find f + g, f - g, fg, and f / g. Determine the domain for each function. f(x) = √x ; g(x) = x - 17.

Learn how to find f + g, f - g, fg, and f / g for the functions f(x) = √x and g(x) = x - 17. This problem involves understanding function operations and determining the domains of these combined functions. Suitable for students in Grades 9-11.

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