Let's go through the problem step by step.
### Part a)
Finding the domains of each function and their combinations:
1. Domain of \( f(x) = x - 1 \):
This function is a linear function and is defined for all real numbers.
[tex]\[
\text{Domain of } f = (-\infty, \infty)
\][/tex]
2. Domain of \( g(x) = \sqrt{x+3} \):
The square root function is defined when the expression inside the square root is non-negative.
[tex]\[
x + 3 \geq 0 \Rightarrow x \geq -3
\][/tex]
[tex]\[
\text{Domain of } g = [-3, \infty)
\][/tex]
3. Domain of \( (f+g)(x) \):
The domain of \( f+g \) is the intersection of the domains of \( f \) and \( g \).
[tex]\[
\text{Domain of } (f+g) = (-\infty, \infty) \cap [-3, \infty) = [-3, \infty)
\][/tex]
4. Domain of \( (f-g)(x) \):
Similar to \( f+g \), the domain of \( f-g \) is also the intersection of the domains of \( f \) and \( g \).
[tex]\[
\text{Domain of } (f-g) = [-3, \infty)
\][/tex]
5. Domain of \( (fg)(x) \):
The domain of \( f \cdot g \) is again the intersection of the domains of \( f \) and \( g \).
[tex]\[
\text{Domain of } (fg) = [-3, \infty)
\][/tex]
6. Domain of \( ff(x) = f(f(x)) \):
We first find the domain of the inner function \( f(x) \) which is \( (-\infty, \infty) \). Since \( f \) is \( x-1 \), applying \( f \) again results in \( f(f(x)) = f(x-1) = (x-1) - 1 = x - 2 \). The function \( x-2 \) is also defined for all real numbers.
[tex]\[
\text{Domain of } ff = (-\infty, \infty)
\][/tex]
7. Domain of \( \frac{f}{g}(x) \):
The function \( g(x) \) must not be zero to avoid division by zero. Since \( g(x) = \sqrt{x+3} \), \( g(x) = 0 \) at \( x = -3 \).
[tex]\[
x \neq -3
\][/tex]
So, the domain is:
[tex]\[
\text{Domain of } \frac{f}{g} = [-3, \infty) \setminus \{-3\}
\][/tex]
8. Domain of \( \frac{g}{f}(x) \):
The function \( f(x) \) must not be zero to avoid division by zero. Since \( f(x) = x - 1 \), \( f(x) = 0 \) at \( x = 1 \).
[tex]\[
x \neq 1
\][/tex]
So, the domain is:
[tex]\[
\text{Domain of } \frac{g}{f} = [-3, \infty) \setminus \{1\}
\][/tex]
### Part b)
Finding the expressions for combined functions:
1. \( (f+g)(x) \):
[tex]\[
(f+g)(x) = f(x) + g(x) = (x-1) + \sqrt{x+3}
\][/tex]
[tex]\[
(f+g)(x) = x - 1 + \sqrt{x+3}
\][/tex]
2. \( (f-g)(x) \):
[tex]\[
(f-g)(x) = f(x) - g(x) = (x-1) - \sqrt{x+3}
\][/tex]
[tex]\[
(f-g)(x) = x - 1 - \sqrt{x+3}
\][/tex]
3. \( (fg)(x) \):
[tex]\[
(fg)(x) = f(x) \cdot g(x) = (x-1) \cdot \sqrt{x+3}
\][/tex]
[tex]\[
(fg)(x) = (x-1) \sqrt{x+3}
\][/tex]
4. \( ff(x) \):
[tex]\[
ff(x) = f(f(x)) = f(x-1) = (x-1) - 1
\][/tex]
[tex]\[
ff(x) = x - 2
\][/tex]
5. \( \left( \frac{f}{g} \right)(x) \):
[tex]\[
\left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} = \frac{x-1}{\sqrt{x+3}}
\][/tex]
6. \( \left( \frac{g}{f} \right)(x) \):
[tex]\[
\left( \frac{g}{f} \right)(x) = \frac{g(x)}{f(x)} = \frac{\sqrt{x+3}}{x-1}
\][/tex]
### Summary of Part a)
- Domain of \( f \): \((-∞, ∞)\)
- Domain of \( g \): \([-3, ∞)\)
- Domain of \( (f+g) \): \([-3, ∞)\)
- Domain of \( (f-g) \): \([-3, ∞)\)
- Domain of \( (fg) \): \([-3, ∞)\)
- Domain of \( ff \): \((-∞, ∞)\)
- Domain of \( \frac{f}{g} \): \([-3, ∞) \setminus \{-3\}\)
- Domain of [tex]\( \frac{g}{f} \)[/tex]: [tex]\([-3, ∞) \setminus \{1\}\)[/tex]
