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\\)

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.)
c. the domain of \\(f-g\\) is \\(\varnothing\\).

\\((fg)(x) = 2x^3 - 25x^2 + 104x - 144\\) (simplify your answer.)

what is the domain of \\(fg\\)?

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

Explanation:

Define the functions and their domains

The given functions are:

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

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

Both \(f(x)\) and \(g(x)\) are polynomials, so their individual domains are all real numbers:

$$ D_f = (-\infty, \infty), \quad D_g = (-\infty, \infty) $$

Determine the domain of the product function fg

The product function is defined as:

$$ (fg)(x) = f(x) \cdot g(x) $$

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

$$ D_{fg} = D_f \cap D_g = (-\infty, \infty) \cap (-\infty, \infty) = (-\infty, \infty) $$

Match with the multiple-choice options

The options for the domain of \(fg\) are:

• A. The domain of \(fg\) is \(\{ \}\).
• B. The domain of \(fg\) is \(\quad\) (Type your answer in interval notation.)
• C. The domain of \(fg\) is \(\varnothing\).

Since the domain is \((-\infty, \infty)\), option B is the correct choice, with the interval \((-\infty, \infty)\) filled in.

Answer:

• (A) The domain of fg is { }
• (B) The domain of fg is \((-\infty, \infty)\) (Correct answer)
• (C) The domain of fg is \(\varnothing\)