Question

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Part 1
Find f plus g​, f minus g​, ​fg, and StartFraction f Over g EndFraction
. Determine the domain for each function.
f left parenthesis x right parenthesis equals StartRoot x minus 6 EndRoot​; g left parenthesis x right parenthesis equals StartRoot x minus 8 EndRoot
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Part 1
​(f plus g​)(x) ​=
  
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Answer 1

To find \( (f + g)(x) \), \( (f - g)(x) \), \( (fg)(x) \), and \( \frac{f}{g}(x) \) for the functions \( f(x) = \sqrt{x - 6} \) and \( g(x) = \sqrt{x - 8} \), let's start with each operation:

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

\[
(f + g)(x) = f(x) + g(x) = \sqrt{x - 6} + \sqrt{x - 8}
\]

### Domain of \( (f + g)(x) \)

To find the domain, we need both square roots to be defined, which means the expressions under the square root must be non-negative:

1. \( x - 6 \geq 0 \) ⟹ \( x \geq 6 \)
2. \( x - 8 \geq 0 \) ⟹ \( x \geq 8 \)

The more restrictive condition is \( x \geq 8 \). Therefore, the domain of \( (f + g)(x) \) is:

**Domain:** \( [8, \infty) \)

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

\[
(f - g)(x) = f(x) - g(x) = \sqrt{x - 6} - \sqrt{x - 8}
\]

### Domain of \( (f - g)(x) \)

Using the same consideration for domains as above, we find:

**Domain:** \( [8, \infty) \)

### 3. \( (fg)(x) \)

\[
(fg)(x) = f(x) \cdot g(x) = \sqrt{x - 6} \cdot \sqrt{x - 8} = \sqrt{(x - 6)(x - 8)}
\]

### Domain of \( (fg)(x) \)

Again, we have the same conditions from before:

**Domain:** \( [8, \infty) \)

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

\[
\frac{f}{g}(x) = \frac{f(x)}{g(x)} = \frac{\sqrt{x - 6}}{\sqrt{x - 8}}
\]

### Domain of \( \frac{f}{g}(x) \)

For the fraction to be defined, \( g(x) \) must not be zero:

1. \( g(x) \geq 0 \) ⟹ \( x \geq 8 \)
2. \( g(x) \neq 0 \) ⟹ \( x - 8 \neq 0 \) ⟹ \( x \neq 8 \)

So, the domain is:

**Domain:** \( (8, \infty) \)

### Summary

1. **\( (f + g)(x) = \sqrt{x - 6} + \sqrt{x - 8} \): Domain** \( [8, \infty) \)
2. **\( (f - g)(x) = \sqrt{x - 6} - \sqrt{x - 8} \): Domain** \( [8, \infty) \)
3. **\( (fg)(x) = \sqrt{(x - 6)(x - 8)} \): Domain** \( [8, \infty) \)
4. **\( \frac{f}{g}(x) = \frac{\sqrt{x - 6}}{\sqrt{x - 8}} \): Domain** \( (8, \infty) \)