To solve the given problem, we need to determine the domains of the functions \( f \) and \( g \), as well as their combinations. Then, we will find the expressions for the combined functions.
- Domain of \( f(x) = x - 5 \): Since \( f(x) \) is a linear function, its domain is all real numbers, \( (-\infty, \infty) \).
- Domain of \( g(x) = \sqrt{x + 6} \): The expression inside the square root must be non-negative, so \( x + 6 \geq 0 \). Therefore, the domain is \( [-6, \infty) \).
For the combined functions:
- \( f + g \): The domain is the intersection of the domains of \( f \) and \( g \), which is \( [-6, \infty) \).
- \( f - g \): The domain is the same as \( f + g \), which is \( [-6, \infty) \).
- \( f \cdot g \): The domain is the same as \( f + g \), which is \( [-6, \infty) \).
- \( \frac{f}{g} \): The domain is \( [-6, \infty) \) excluding points where \( g(x) = 0 \). Since \( g(x) = \sqrt{x + 6} \), \( g(x) = 0 \) when \( x = -6 \). Thus, the domain is \( (-6, \infty) \).
- \( \frac{g}{f} \): The domain is \( [-6, \infty) \) excluding points where \( f(x) = 0 \). Since \( f(x) = x - 5 \), \( f(x) = 0 \) when \( x = 5 \). Thus, the domain is \( [-6, 5) \cup (5, \infty) \).
- \( (f + g)(x) \): This is \( f(x) + g(x) \).
- \( (f - g)(x) \): This is \( f(x) - g(x) \).
- \( (f \cdot g)(x) \): This is \( f(x) \cdot g(x) \).
- \( \left(\frac{f}{g}\right)(x) \): This is \( \frac{f(x)}{g(x)} \).
- \( \left(\frac{g}{f}\right)(x) \): This is \( \frac{g(x)}{f(x)} \).
The function \( f(x) = x - 5 \) is a linear function. Linear functions are defined for all real numbers.
\[
\text{Domain of } f: (-\infty, \infty)
\]
The function \( g(x) = \sqrt{x + 6} \) is defined when the expression inside the square root is non-negative.
\[
x + 6 \geq 0 \implies x \geq -6
\]
\[
\text{Domain of } g: [-6, \infty)
\]
- \( f + g \): The domain is the intersection of the domains of \( f \) and \( g \).
\[
\text{Domain of } f + g: [-6, \infty)
\]
- \( f - g \): The domain is the same as \( f + g \).
\[
\text{Domain of } f - g: [-6, \infty)
\]
- \( f \cdot g \): The domain is the same as \( f + g \).
\[
\text{Domain of } f \cdot g: [-6, \infty)
\]
- \( \frac{f}{g} \): The domain is \( [-6, \infty) \) excluding points where \( g(x) = 0 \). Since \( g(x) = \sqrt{x + 6} \), \( g(x) = 0 \) when \( x = -6 \).
\[
\text{Domain of } \frac{f}{g}: (-6, \infty)
\]
- \( \frac{g}{f} \): The domain is \( [-6, \infty) \) excluding points where \( f(x) = 0 \). Since \( f(x) = x - 5 \), \( f(x) = 0 \) when \( x = 5 \).
\[
\text{Domain of } \frac{g}{f}: [-6, 5) \cup (5, \infty)
\]
\[
(f + g)(x) = f(x) + g(x) = x - 5 + \sqrt{x + 6}
\]
\[
(f - g)(x) = f(x) - g(x) = x - 5 - \sqrt{x + 6}
\]
\[
(f \cdot g)(x) = f(x) \cdot g(x) = (x - 5) \sqrt{x + 6}
\]
- \( \left(\frac{f}{g}\right)(x) \):
\[
\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{x - 5}{\sqrt{x + 6}}
\]
- \( \left(\frac{g}{f}\right)(x) \):
\[
\left(\frac{g}{f}\right)(x) = \frac{g(x)}{f(x)} = \frac{\sqrt{x + 6}}{x - 5}
\]
\[
\boxed{[-6, \infty)}
\]
Answered
