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Answer :
To find the function that represents the profit of the company after [tex]\( x \)[/tex] years, we need to subtract the expenses from the revenue.
Let's break down the problem step-by-step:
1. Revenue Function [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = 2x^4 + x^3 - 4x + 20
\][/tex]
This represents the company's revenue in thousands of dollars after [tex]\( x \)[/tex] years.
2. Expenses Function [tex]\( g(x) \)[/tex]:
[tex]\[
g(x) = -x^3 + x^2 + 50
\][/tex]
This represents the company's expenses in thousands of dollars after [tex]\( x \)[/tex] years.
3. Profit Function:
The profit function [tex]\( P(x) \)[/tex] is the revenue function minus the expenses function:
[tex]\[
P(x) = f(x) - g(x)
\][/tex]
Substituting the expressions for [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[
P(x) = (2x^4 + x^3 - 4x + 20) - (-x^3 + x^2 + 50)
\][/tex]
4. Simplifying the Profit Function:
Distribute the negative sign through the terms of [tex]\( g(x) \)[/tex]:
[tex]\[
P(x) = 2x^4 + x^3 - 4x + 20 + x^3 - x^2 - 50
\][/tex]
Combine like terms:
- For [tex]\( x^4 \)[/tex]: [tex]\( 2x^4 \)[/tex]
- For [tex]\( x^3 \)[/tex]: [tex]\( x^3 + x^3 = 2x^3 \)[/tex]
- For [tex]\( x^2 \)[/tex]: [tex]\(-x^2 \)[/tex]
- For [tex]\( x \)[/tex]: [tex]\(-4x \)[/tex]
- Constant terms: [tex]\( 20 - 50 = -30 \)[/tex]
The simplified profit function is:
[tex]\[
P(x) = 2x^4 + 2x^3 - x^2 - 4x - 30
\][/tex]
5. Domain of the Profit Function:
Polynomial functions are defined for all real numbers unless there are specific restrictions given by the context. Here, no restrictions are specified, so the domain of the profit function is all real numbers:
[tex]\[
(-\infty, \infty)
\][/tex]
Therefore, the correct function representing the profit and its domain is:
[tex]\[
P(x) = 2x^4 + 2x^3 - x^2 - 4x - 30 \quad \text{and the domain is} \quad (-\infty, \infty)
\][/tex]
This matches the option: [tex]\((f-g)(x)=2x^4+2x^3-x^2-4x-30\)[/tex] and the domain is [tex]\((-\infty, \infty)\)[/tex].
Let's break down the problem step-by-step:
1. Revenue Function [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = 2x^4 + x^3 - 4x + 20
\][/tex]
This represents the company's revenue in thousands of dollars after [tex]\( x \)[/tex] years.
2. Expenses Function [tex]\( g(x) \)[/tex]:
[tex]\[
g(x) = -x^3 + x^2 + 50
\][/tex]
This represents the company's expenses in thousands of dollars after [tex]\( x \)[/tex] years.
3. Profit Function:
The profit function [tex]\( P(x) \)[/tex] is the revenue function minus the expenses function:
[tex]\[
P(x) = f(x) - g(x)
\][/tex]
Substituting the expressions for [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[
P(x) = (2x^4 + x^3 - 4x + 20) - (-x^3 + x^2 + 50)
\][/tex]
4. Simplifying the Profit Function:
Distribute the negative sign through the terms of [tex]\( g(x) \)[/tex]:
[tex]\[
P(x) = 2x^4 + x^3 - 4x + 20 + x^3 - x^2 - 50
\][/tex]
Combine like terms:
- For [tex]\( x^4 \)[/tex]: [tex]\( 2x^4 \)[/tex]
- For [tex]\( x^3 \)[/tex]: [tex]\( x^3 + x^3 = 2x^3 \)[/tex]
- For [tex]\( x^2 \)[/tex]: [tex]\(-x^2 \)[/tex]
- For [tex]\( x \)[/tex]: [tex]\(-4x \)[/tex]
- Constant terms: [tex]\( 20 - 50 = -30 \)[/tex]
The simplified profit function is:
[tex]\[
P(x) = 2x^4 + 2x^3 - x^2 - 4x - 30
\][/tex]
5. Domain of the Profit Function:
Polynomial functions are defined for all real numbers unless there are specific restrictions given by the context. Here, no restrictions are specified, so the domain of the profit function is all real numbers:
[tex]\[
(-\infty, \infty)
\][/tex]
Therefore, the correct function representing the profit and its domain is:
[tex]\[
P(x) = 2x^4 + 2x^3 - x^2 - 4x - 30 \quad \text{and the domain is} \quad (-\infty, \infty)
\][/tex]
This matches the option: [tex]\((f-g)(x)=2x^4+2x^3-x^2-4x-30\)[/tex] and the domain is [tex]\((-\infty, \infty)\)[/tex].
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