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+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 (−∞,∞)
.
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 .

(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 .

Answer 1

To find the profit of the company after \( x \) years, we need to determine the profit \( P(x) \) by subtracting the expenses \( g(x) \) from the revenue \( f(x) \).

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

The profit function \( P(x) \) is defined as:

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

Now, let's compute \( P(x) \):

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

Distributing the negative sign:

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

Combining like terms:

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

Now we have the profit function:

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

Next, we analyze the domain. Since \( x \) represents the number of years after starting the company and cannot be negative, the domain is:

\[
x \geq 0
\]

Thus, the final answer is:

**Profit function**: \( P(x) = 2x^4 + 2x^3 - x^2 - 4x - 30 \)

**Domain**: \( x \geq 0 \)

So, the correct response is:

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