Question

find \\(f + g\\), \\(f - g\\), \\(fg\\) and \\(\frac{f}{g}\\). determine the domain for each function.

\\(f(x) = 2x^2 - 17x + 36\\), \\(g(x) = x - 4\\)

\\((f + g)(x) = 2x^2 - 16x + 32\\) (simplify your answer.)

what is the domain of \\(f + g\\)?

a. the domain of \\(f + g\\) is \\(\\{\quad\\}\\). (use a comma to separate answers as needed.)
b. the domain of \\(f + g\\) is \\(\quad\\). (type your answer in interval notation.)
c. the domain of \\(f + g\\) is \\(\varnothing\\).

Explanation:

Find the domain of the individual functions

The given functions are:

$$f(x) = 2x^2 - 17x + 36$$

$$g(x) = x - 4$$

Both \(f(x)\) and \(g(x)\) are polynomial functions. The domain of any polynomial function is all real numbers:

$$\text{Domain}(f) = (-\infty, \infty)$$

$$\text{Domain}(g) = (-\infty, \infty)$$

Determine the domain of the sum function

The sum function is defined as:

$$(f+g)(x) = f(x) + g(x) = 2x^2 - 16x + 32$$

The domain of the sum of two functions is the intersection of their individual domains:

$$\text{Domain}(f+g) = \text{Domain}(f) \cap \text{Domain}(g) = (-\infty, \infty) \cap (-\infty, \infty) = (-\infty, \infty)$$

Match with the given options

Option B states: "The domain of \(f + g\) is \(\quad\) (Type your answer in interval notation.)"
The correct interval notation for all real numbers is \((-\infty, \infty)\).

Answer:

• A. The domain of \(f + g\) is \(\{ \quad \}\). (Use a comma to separate answers as needed.)
• B. The domain of \(f + g\) is \((-\infty, \infty)\). (Type your answer in interval notation.) (Correct answer)
• C. The domain of \(f + g\) is \(\varnothing\).