Questions: Find f+g, f-g, fg and f/g. Determine the domain for each function. f(x)=2x^2+13x+20, g(x)=x+4 (f+g)(x)=2x^2+14x+24 (Simplify your answer.) What is the domain of f+g ? A. The domain of f+g is . (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 ∅.

Solution Steps

To solve the problem, we need to perform the following steps:

1. Addition of Functions: Add the functions $f(x)$ and $g(x)$ to get $(f+g)(x)$.
2. Subtraction of Functions: Subtract $g(x)$ from $f(x)$ to get $(f-g)(x)$.
3. Multiplication of Functions: Multiply $f(x)$ and $g(x)$ to get $(f \cdot g)(x)$.
4. Division of Functions: Divide $f(x)$ by $g(x)$ to get $\left(\frac{f}{g}\right)(x)$.
5. Determine Domains: For each resulting function, determine the domain. The domain of a function is the set of all possible input values (x-values) that do not lead to undefined expressions. For rational functions, this means ensuring the denominator is not zero.

We start by adding the functions $f(x)$ and $g(x)$: $$f(x) + g(x) = (2x^2 + 13x + 20) + (x + 4) = 2x^2 + 14x + 24$$

Next, we subtract $g(x)$ from $f(x)$: $$f(x) - g(x) = (2x^2 + 13x + 20) - (x + 4) = 2x^2 + 12x + 16$$

We then multiply the functions $f(x)$ and $g(x)$: $$f(x) \cdot g(x) = (2x^2 + 13x + 20)(x + 4)$$

Next, we divide $f(x)$ by $g(x)$: $$\frac{f(x)}{g(x)} = \frac{2x^2 + 13x + 20}{x + 4} = 2x + 5$$

The domains for the resulting functions are as follows:

• The domain of $f+g$ is $(-\infty, \infty)$.
• The domain of $f-g$ is $(-\infty, \infty)$.
• The domain of $f \cdot g$ is $(-\infty, \infty)$.
• The domain of $\frac{f}{g}$ is $(-\infty, -4) \cup (-4, \infty)$ since $g(x)$ cannot be zero.

Final Answer