Question

The revenue of a company in thousands of dollars x years after starting is represented by the function f(x)=2x4+x3−4x+20
. The expenses of the company in thousands of dollars x years after starting is represented by the function g(x)=−x3+x2+50
. What function represents the profit of the company after x years? What is the domain?(1 point)
Responses

(f−g)(x)=2x4+2x3−x2−4x−30
and the domain is x≥0
.
open paren f minus g close paren of x is equal to 2 x to the 4th power plus 2 x cubed minus x squared minus 4 x minus 30 and the domain is x is greater than or equal to 0 .

(f+g)(x)=2x4+x2−4x+70
and the domain is x≥0
.
open paren f plus g close paren of x is equal to 2 x to the 4th power plus x squared minus 4 x plus 70 and the domain is x is greater than or equal to 0 .

(f+g)(x)=2x4+x2−4x+70
and the domain is (−∞,∞)
.
open paren f plus g close paren of x is equal to 2 x to the 4th power plus x squared minus 4 x plus 70 and the domain is open paren negative infinity comma infinity close paren .

(f−g)(x)=2x4+2x3−x2−4x−30
and the domain is (−∞,∞)
.

Answer 1

To find the profit function of the company, we need to calculate the difference between the revenue function \( f(x) \) and the expense function \( g(x) \). The profit function \( P(x) \) is given by:

\[
P(x) = f(x) - g(x)
\]

Given the functions:

\[
f(x) = 2x^4 + x^3 - 4x + 20
\]
\[
g(x) = -x^3 + x^2 + 50
\]

We compute \( P(x) \):

\[
P(x) = (2x^4 + x^3 - 4x + 20) - (-x^3 + x^2 + 50)
\]

Distributing the negative sign for \( g(x) \):

\[
P(x) = 2x^4 + x^3 - 4x + 20 + x^3 - x^2 - 50
\]

Combining like terms:

\[
P(x) = 2x^4 + (x^3 + x^3) - x^2 - 4x + (20 - 50)
\]
\[
P(x) = 2x^4 + 2x^3 - x^2 - 4x - 30
\]

Thus, the profit function is:

\[
P(x) = 2x^4 + 2x^3 - x^2 - 4x - 30
\]

Regarding the domain, since both the revenue and expenses are defined for \( x \geq 0 \) (as they represent years after starting the company), the domain of the profit function is also:

\[
\text{Domain: } x \geq 0
\]

So the correct response is:

\((f - g)(x) = 2x^4 + 2x^3 - x^2 - 4x - 30\) and the domain is \( x \geq 0 \).