Find $$ f+g, f-g, f g $$ and $$ \frac{f}{g} $$. Determine the domain for each function. $$ f(x)=5 x+2, g(x)=x-6 $$ c. The domain of $$ f+g $$ is $$ \varnothing $$. $$ (\mathrm{f}-\mathrm{g})(\mathrm{x})=4 \mathrm{x}+8 $$ (Simplify your answer.) What is the domain of $$ f-g $$ ? A. The domain of $$ f-g $$ is $$ \square $$ \} . (Use a comma to separate answers as needed.) B. The domain of $$ f-g $$ is $$ \square $$ . (Type your answer in interval notation.) C. The domain of $$ f-g $$ is $$ \varnothing $$.

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UpStudy AI · Accepted Answer

To determine the domain of the function $$ f - g $$, let's first simplify the expression and then analyze any restrictions.

**Given:**
$$
f(x) = 5x + 2 \
g(x) = x - 6
$$

**Simplify $$ f - g $$:**
$$
(f - g)(x) = f(x) - g(x) = (5x + 2) - (x - 6) = 5x + 2 - x + 6 = 4x + 8
$$

**Determine the Domain:**

Both $$ f(x) $$ and $$ g(x) $$ are linear functions, which are defined for all real numbers. When you subtract two linear functions, the resulting function $$ 4x + 8 $$ is also a linear function and thus defined for all real numbers.

**Therefore:**

- **A.** The domain of $$ f - g $$ is $$ \mathbb{R} $$.  
  *(If you need to enter multiple answers separated by commas, it would simply be redundant since it's all real numbers.)*

- **B.** The domain of $$ f - g $$ in interval notation is $$ (-\infty, \infty) $$.

- **C.** The domain of $$ f - g $$ is **not** $$ \varnothing $$ (the empty set), because $$ f - g $$ is defined for all real numbers.

**Correct Choice:**

**B.** The domain of $$ f - g $$ is $$ (-\infty, \infty) $$.
The domain of $$ f - g $$ is all real numbers, which in interval notation is $$ (-\infty, \infty) $$.

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