Find $$ f+g, f-g, f g $$ and $$ \frac{f}{g} $$. Determine the domain for each function. $$ f(x)=5 x+8, g(x)=x-3 $$ ( $$ f+g)(x)=\square $$ (Simplify your answer.) What is the domain of $$ f+g $$ ? A. The domain of $$ f+g $$ is $$ \} $$. (Use a comma to separate answers as needed.) B. The domain of $$ f+g $$ is C. The domain of $$ f+g $$ is $$ \varnothing $$. $$ T y $$. ( $$ f-g)(x)=\square $$ (Simplify your answer.) What is the domain of $$ f-g $$ ? A. The domain of interval notation.) $$ f $$. $$ g $$ is

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$$ f(x)=5x+8 $$ and $$ g(x)=x-3 $$.

1. **Sum:**

$$
   (f+g)(x)=f(x)+g(x)=(5x+8)+(x-3)=6x+5
   $$
   
   **Domain:** Both $$ f(x) $$ and $$ g(x) $$ are defined for all real numbers, so the domain is  
   $$
   (-\infty,\infty).
   $$

2. **Difference:**

$$
   (f-g)(x)=f(x)-g(x)=(5x+8)-(x-3)=4x+11
   $$
   
   **Domain:** Since both functions are defined for all real numbers, the domain is  
   $$
   (-\infty,\infty).
   $$

3. **Product:**

$$
   (fg)(x)=f(x) \cdot g(x)=(5x+8)(x-3)
   $$
   
   Expand the product:
   
   $$
   5x(x-3)+8(x-3)=5x^2-15x+8x-24=5x^2-7x-24
   $$
   
   **Domain:** All real numbers, so the domain is  
   $$
   (-\infty,\infty).
   $$

4. **Quotient:**

$$
   \left(\frac{f}{g}\right)(x)=\frac{5x+8}{x-3}
   $$
   
   **Domain:** The denominator cannot be zero, so set:
   
   $$
   x-3 \neq 0 \quad\Longrightarrow\quad x \neq 3.
   $$
   
   Therefore, the domain is  
   $$
   (-\infty,3) \cup (3,\infty).
   $$
$$ (f+g)(x) = 6x + 5 $$ Domain of $$ f+g $$: All real numbers, $$ (-\infty, \infty) $$ $$ (f-g)(x) = 4x + 11 $$ Domain of $$ f-g $$: All real numbers, $$ (-\infty, \infty) $$ $$ fg(x) = 5x^2 - 7x - 24 $$ Domain of $$ fg $$: All real numbers, $$ (-\infty, \infty) $$ $$ \frac{f}{g}(x) = \frac{5x+8}{x-3} $$ Domain of $$ \frac{f}{g} $$: All real numbers except $$ x = 3 $$, $$ (-\infty, 3) \cup (3, \infty) $$

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