The revenue of a company in thousands of dollars x years after starting is represented by the function f(x)=2x4+x3−4x+20

Question

. 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 (−∞,∞)
.
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+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+2x3−x2−4x−30
and the domain is (−∞,∞)
.
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 open paren negative infinity comma infinity close paren .

Answer 1

To find the profit function \( P(x) \), we need to subtract the expenses \( g(x) \) from the revenue \( f(x) \).

The functions given are:

• Revenue function: \( f(x) = 2x^4 + x^3 - 4x + 20 \)
• Expense function: \( g(x) = -x^3 + x^2 + 50 \)

The profit function \( P(x) \) is given by:

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

Substituting in the values for \( f(x) \) and \( g(x) \):

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

Distributing the negative sign:

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

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:

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

Now, regarding the domain, since \( x \) represents years after starting the company, \( x \) must be non-negative. Therefore, the domain is:

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

In conclusion, the correct response is:

\[ \boxed{(f-g)(x) = 2x^4 + 2x^3 - x^2 - 4x - 30 \text{ and the domain is } x \geq 0.} \]