The functions $ f $ and $ g $ are defined as $ f(x) = x - 1 $ and $ g(x) = \sqrt{x + 3} $.

a) Find the domain of $ f $, $ g $, $ f + g $, $ f - g $, $ f g $, $ f \circ f $, $ \frac{f}{g} $, and $ \frac{g}{f} $.

b) Find $ (f + g)(x) $, $ (f - g)(x) $, $ (fg)(x) $, $ (f \circ f)(x) $, $ \left(\frac{f}{g}\right)(x) $, and $ \left(\frac{g}{f}\right)(x) $.

a) The domain of $ f $ is $\square$
(Type your answer in interval notation.)

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]