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