Find $$ f+g, f-g $$,fg and $$ \frac{f}{g} $$. Determine the domain for each function. $$ f(x)=5 x-3, g(x)=x-1 $$ $$ (f+g)(x)=\square $$ (Simplify your answer.)

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1. Compute $$ f+g $$:
   - Write the sum:  
     $$
     (f+g)(x)=f(x)+g(x)=(5x-3)+(x-1).
     $$
   - Combine like terms:
     $$
     (5x + x) - (3+1)=6x-4.
     $$
   - Thus, 
     $$
     (f+g)(x)=6x-4.
     $$
   - Domain: Since both $$ f(x) $$ and $$ g(x) $$ are defined for all real numbers, the domain is $$ \mathbb{R} $$.

2. Compute $$ f-g $$:
   - Write the difference:
     $$
     (f-g)(x)=f(x)-g(x)=(5x-3)-(x-1).
     $$
   - Distribute the negative sign and combine like terms:
     $$
     5x-3-x+1=4x-2.
     $$
   - Thus, 
     $$
     (f-g)(x)=4x-2.
     $$
   - Domain: Both $$ f(x) $$ and $$ g(x) $$ are defined for all $$ x\in\mathbb{R} $$, so the domain is $$ \mathbb{R} $$.

3. Compute $$ fg $$:
   - Multiply the functions:
     $$
     (fg)(x)=f(x)\cdot g(x)=(5x-3)(x-1).
     $$
   - Expand the product:
     $$
     5x\cdot x-5x\cdot 1-3\cdot x+3\cdot 1=5x^2-5x-3x+3.
     $$
   - Combine like terms:
     $$
     5x^2-8x+3.
     $$
   - Thus,
     $$
     (fg)(x)=5x^2-8x+3.
     $$
   - Domain: The domain is $$ \mathbb{R} $$.

4. Compute $$ \frac{f}{g} $$:
   - Write the quotient:
     $$
     \left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)}=\frac{5x-3}{x-1}.
     $$
   - Determine the domain: The expression is undefined when the denominator is zero. Solve
     $$
     x-1=0 \quad \Rightarrow \quad x=1.
     $$
   - Thus, the domain is:
     $$
     \{x\in\mathbb{R}\mid x\neq 1\}.
     $$

Summary:

- $$\boxed{(f+g)(x)=6x-4, \quad \text{domain: } \mathbb{R}}$$
- $$\boxed{(f-g)(x)=4x-2, \quad \text{domain: } \mathbb{R}}$$
- $$\boxed{(fg)(x)=5x^2-8x+3, \quad \text{domain: } \mathbb{R}}$$
- $$\boxed{\left(\frac{f}{g}\right)(x)=\frac{5x-3}{x-1}, \quad \text{domain: } \{x\in\mathbb{R}\mid x\neq 1\}}$$
$$ (f+g)(x) = 6x - 4, \quad \text{domain: all real numbers} $$ $$ (f-g)(x) = 4x - 2, \quad \text{domain: all real numbers} $$ $$ (fg)(x) = 5x^2 - 8x + 3, \quad \text{domain: all real numbers} $$ $$ \left(\frac{f}{g}\right)(x) = \frac{5x - 3}{x - 1}, \quad \text{domain: all real numbers except } x = 1 $$

Find ,fg and . Determine the domain for each function.

(Simplify your answer.)

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Find ,fg and . Determine the domain for each function. (Simplify your answer.)