If F ( X ) = 2 X 2 + 1 F(x) = 2x^2 + 1 F ( X ) = 2 X 2 + 1 And G ( X ) = X 2 − 7 G(x) = X^2 - 7 G ( X ) = X 2 − 7 , Find ( F − G ) ( X (f-g)(x ( F − G ) ( X ].A. X 2 + 8 X^2 + 8 X 2 + 8 B. X 2 − 6 X^2 - 6 X 2 − 6 C. 3 X 2 − 6 3x^2 - 6 3 X 2 − 6 D. 3 X 2 + 8 3x^2 + 8 3 X 2 + 8

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Understanding the Problem

To find the difference between two functions, we need to subtract the second function from the first function. In this case, we are given two functions: $f(x) = 2x^2 + 1$ and $g(x) = x^2 - 7$. We need to find the difference between these two functions, which is denoted by $(f-g)(x)$.

Subtracting the Functions

To find the difference between the two functions, we need to subtract the second function from the first function. This means that we need to subtract $g(x)$ from $f(x)$. We can do this by subtracting the corresponding terms in the two functions.

Step 1: Subtract the $x^2$ Terms

The first term in both functions is $x^2$. To subtract the $x^2$ terms, we need to subtract the coefficient of the $x^2$ term in the second function from the coefficient of the $x^2$ term in the first function. In this case, the coefficient of the $x^2$ term in the first function is 2, and the coefficient of the $x^2$ term in the second function is 1. Therefore, we need to subtract 1 from 2, which gives us 1.

Step 2: Subtract the Constant Terms

The next term in both functions is a constant term. To subtract the constant terms, we need to subtract the constant term in the second function from the constant term in the first function. In this case, the constant term in the first function is 1, and the constant term in the second function is -7. Therefore, we need to subtract -7 from 1, which gives us 8.

Combining the Terms

Now that we have subtracted the $x^2$ terms and the constant terms, we can combine the terms to find the difference between the two functions. The difference between the two functions is given by:

$(f-g)(x) = (2x^2 + 1) - (x^2 - 7)$

$= 2x^2 + 1 - x^2 + 7$

$= x^2 + 8$

Conclusion

Therefore, the difference between the two functions is $x^2 + 8$. This is the correct answer.

Answer Key

The correct answer is A. $x^2 + 8$.

Explanation

To find the difference between two functions, we need to subtract the second function from the first function. In this case, we are given two functions: $f(x) = 2x^2 + 1$ and $g(x) = x^2 - 7$. We need to find the difference between these two functions, which is denoted by $(f-g)(x)$. To find the difference, we need to subtract the corresponding terms in the two functions. This means that we need to subtract the $x^2$ terms and the constant terms. After subtracting the terms, we can combine the terms to find the difference between the two functions. The difference between the two functions is given by $(f-g)(x) = x^2 + 8$.

Example

Let's consider an example to illustrate the concept of finding the difference between two functions. Suppose we have two functions: $f(x) = 3x^2 + 2$ and $g(x) = 2x^2 - 1$. We need to find the difference between these two functions, which is denoted by $(f-g)(x)$. To find the difference, we need to subtract the corresponding terms in the two functions. This means that we need to subtract the $x^2$ terms and the constant terms. After subtracting the terms, we can combine the terms to find the difference between the two functions. The difference between the two functions is given by $(f-g)(x) = 3x^2 + 2 - 2x^2 + 1 = x^2 + 3$.

Real-World Applications

The concept of finding the difference between two functions has many real-world applications. For example, in economics, the difference between two functions can be used to model the difference between two economic variables, such as the difference between the demand and supply of a product. In physics, the difference between two functions can be used to model the difference between two physical quantities, such as the difference between the kinetic energy and potential energy of an object.

Conclusion

In conclusion, finding the difference between two functions is an important concept in mathematics that has many real-world applications. To find the difference between two functions, we need to subtract the second function from the first function. This means that we need to subtract the corresponding terms in the two functions. After subtracting the terms, we can combine the terms to find the difference between the two functions. The difference between the two functions is given by $(f-g)(x) = x^2 + 8$.

Q: What is the difference between two functions?

A: The difference between two functions is the result of subtracting one function from another. It is denoted by $(f-g)(x)$, where $f(x)$ and $g(x)$ are the two functions being subtracted.

Q: How do I find the difference between two functions?

A: To find the difference between two functions, you need to subtract the second function from the first function. This means that you need to subtract the corresponding terms in the two functions. After subtracting the terms, you can combine the terms to find the difference between the two functions.

Q: What are the steps to find the difference between two functions?

A: The steps to find the difference between two functions are as follows:

1. Subtract the $x^2$ terms: Subtract the coefficient of the $x^2$ term in the second function from the coefficient of the $x^2$ term in the first function.
2. Subtract the constant terms: Subtract the constant term in the second function from the constant term in the first function.
3. Combine the terms: Combine the terms to find the difference between the two functions.

Q: What is the formula for finding the difference between two functions?

A: The formula for finding the difference between two functions is:

$(f-g)(x) = (a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0) - (b_nx^n + b_{n-1}x^{n-1} + \ldots + b_1x + b_0)$

where $a_n, a_{n-1}, \ldots, a_1, a_0$ are the coefficients of the first function, and $b_n, b_{n-1}, \ldots, b_1, b_0$ are the coefficients of the second function.

Q: Can I use the formula to find the difference between two functions with different degrees?

A: Yes, you can use the formula to find the difference between two functions with different degrees. However, you need to make sure that the terms with the same degree are subtracted correctly.

Q: What are some real-world applications of finding the difference between two functions?

A: Some real-world applications of finding the difference between two functions include:

• Modeling the difference between two economic variables, such as the difference between the demand and supply of a product.
• Modeling the difference between two physical quantities, such as the difference between the kinetic energy and potential energy of an object.
• Finding the difference between two functions in order to determine the maximum or minimum value of a function.

Q: Can I use technology to find the difference between two functions?

A: Yes, you can use technology to find the difference between two functions. Many graphing calculators and computer algebra systems can be used to find the difference between two functions.

Q: What are some common mistakes to avoid when finding the difference between two functions?

A: Some common mistakes to avoid when finding the difference between two functions include:

• Subtracting the terms incorrectly.
• Not combining the terms correctly.
• Not checking the degree of the terms.
• Not using the correct formula.

Q: Can I find the difference between two functions with complex coefficients?

A: Yes, you can find the difference between two functions with complex coefficients. However, you need to make sure that the terms with the same degree are subtracted correctly.

Q: What are some tips for finding the difference between two functions?

A: Some tips for finding the difference between two functions include:

• Make sure to subtract the terms correctly.
• Combine the terms correctly.
• Check the degree of the terms.
• Use the correct formula.
• Check your work carefully.

Q: Can I find the difference between two functions with rational coefficients?

A: Yes, you can find the difference between two functions with rational coefficients. However, you need to make sure that the terms with the same degree are subtracted correctly.

Q: What are some common applications of finding the difference between two functions in science and engineering?

A: Some common applications of finding the difference between two functions in science and engineering include:

• Modeling the difference between two physical quantities, such as the difference between the kinetic energy and potential energy of an object.
• Finding the difference between two functions in order to determine the maximum or minimum value of a function.
• Modeling the difference between two economic variables, such as the difference between the demand and supply of a product.

Q: Can I find the difference between two functions with trigonometric coefficients?

A: Yes, you can find the difference between two functions with trigonometric coefficients. However, you need to make sure that the terms with the same degree are subtracted correctly.

Q: What are some common mistakes to avoid when finding the difference between two functions with trigonometric coefficients?

A: Some common mistakes to avoid when finding the difference between two functions with trigonometric coefficients include:

• Subtracting the terms incorrectly.
• Not combining the terms correctly.
• Not checking the degree of the terms.
• Not using the correct formula.

Q: Can I find the difference between two functions with exponential coefficients?

A: Yes, you can find the difference between two functions with exponential coefficients. However, you need to make sure that the terms with the same degree are subtracted correctly.

Q: What are some common applications of finding the difference between two functions in economics?

A: Some common applications of finding the difference between two functions in economics include:

• Modeling the difference between two economic variables, such as the difference between the demand and supply of a product.
• Finding the difference between two functions in order to determine the maximum or minimum value of a function.
• Modeling the difference between two physical quantities, such as the difference between the kinetic energy and potential energy of an object.

Q: Can I find the difference between two functions with logarithmic coefficients?

A: Yes, you can find the difference between two functions with logarithmic coefficients. However, you need to make sure that the terms with the same degree are subtracted correctly.

Q: What are some common mistakes to avoid when finding the difference between two functions with logarithmic coefficients?

A: Some common mistakes to avoid when finding the difference between two functions with logarithmic coefficients include:

• Subtracting the terms incorrectly.
• Not combining the terms correctly.
• Not checking the degree of the terms.
• Not using the correct formula.