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

Question

UpStudy AI · Accepted Answer

To determine the domain of the function $$ f + g $$, let's break down the problem step by step.

### Given Functions:
- $$ f(x) = 5x + 2 $$
- $$ g(x) = x - 6 $$

### Finding $$ f + g $$:
$$
(f + g)(x) = f(x) + g(x) = (5x + 2) + (x - 6) = 6x - 4
$$

### Determining the Domain of $$ f + g $$:

Both $$ f(x) $$ and $$ g(x) $$ are linear functions. Linear functions are defined for all real numbers because there are no restrictions (like division by zero or square roots of negative numbers) that would limit their domains.

Therefore, the domain of $$ f + g $$ is all real numbers.

#### Answer Choice A:
$$
\text{A. The domain of } f + g \text{ is } \boxed{\,(-\infty, \infty)\,}.
$$

#### Reasons Why Other Options Are Incorrect:
- **B.** This option suggests listing specific values, which isn't applicable since the domain includes all real numbers.
- **C.** The empty set $$ \varnothing $$ would imply that there are no valid inputs for $$ f + g $$, which is not the case here.

### Final Answer:
**A. The domain of $$ f+g $$ is $$ (-\infty, \infty) $$.**
The domain of $$ f+g $$ is all real numbers, which is $$ (-\infty, \infty) $$.

Solución de tutoría real

Responder

Solución

Extra Insights

preguntas relacionadas

Latest Algebra Questions