If $f(x) = 4x^2 + 1$ And $g(x) = X^2 - 5$, Find $(f-g)(x$\].A. $5x^2 - 6$B. $3x^2 + 6$C. $3x^2 - 4$D. $5x^2 - 4$

If $f(x) = 4x^2 + 1$ and $g(x) = x^2 - 5$, find $(f-g)(x)$

Understanding the Problem

In this problem, we are given two functions, $f(x)$ and $g(x)$, and we are asked to find the difference between them, denoted as $(f-g)(x)$. To do this, we need to understand the concept of function subtraction and how it is applied.

Function Subtraction

Function subtraction is a way of combining two functions to create a new function. It is denoted as $(f-g)(x)$, where $f(x)$ and $g(x)$ are the two functions being subtracted. When we subtract two functions, we are essentially finding the difference between their outputs for a given input.

Step 1: Write Down the Functions

The first step in finding $(f-g)(x)$ is to write down the two functions, $f(x)$ and $g(x)$.

$f(x) = 4x^2 + 1$

$g(x) = x^2 - 5$

Step 2: Subtract the Functions

Now that we have written down the two functions, we can subtract them to find $(f-g)(x)$. To do this, we need to subtract the corresponding terms in the two functions.

$(f-g)(x) = (4x^2 + 1) - (x^2 - 5)$

Step 3: Simplify the Expression

Now that we have subtracted the two functions, we can simplify the expression to find the final answer.

$(f-g)(x) = 4x^2 + 1 - x^2 + 5$

Combining Like Terms

When we simplify the expression, we can combine like terms to get the final answer.

$(f-g)(x) = 3x^2 + 6$

Conclusion

In this problem, we were given two functions, $f(x)$ and $g(x)$, and we were asked to find the difference between them, denoted as $(f-g)(x)$. We used the concept of function subtraction to find the final answer, which is $3x^2 + 6$.

Answer

The final answer is $\boxed{3x^2 + 6}$.

Explanation

The final answer is $3x^2 + 6$ because when we subtract the two functions, we get $4x^2 + 1 - x^2 + 5$, which simplifies to $3x^2 + 6$.

Key Concepts

• Function subtraction
• Combining like terms
• Simplifying expressions

Real-World Applications

Function subtraction has many real-world applications, such as:

• Finding the difference between two financial functions
• Calculating the difference between two physical quantities
• Determining the difference between two chemical reactions

Practice Problems

Here are some practice problems to help you understand function subtraction:

1. If $f(x) = 2x^2 + 3$ and $g(x) = x^2 - 2$, find $(f-g)(x)$.
2. If $f(x) = x^2 + 4$ and $g(x) = 2x^2 - 3$, find $(f-g)(x)$.
3. If $f(x) = 3x^2 - 2$ and $g(x) = x^2 + 1$, find $(f-g)(x)$.

Solutions

Here are the solutions to the practice problems:

1. $(f-g)(x) = 2x^2 + 3 - (x^2 - 2) = x^2 + 5$
2. $(f-g)(x) = x^2 + 4 - (2x^2 - 3) = -x^2 + 7$
3. $(f-g)(x) = 3x^2 - 2 - (x^2 + 1) = 2x^2 - 3$

Conclusion

In this article, we learned how to find the difference between two functions using function subtraction. We used the concept of function subtraction to find the final answer, which is $3x^2 + 6$. We also discussed the key concepts of function subtraction, combining like terms, and simplifying expressions. Finally, we provided some practice problems to help you understand function subtraction.
Q&A: Function Subtraction

Understanding Function Subtraction

In our previous article, we discussed how to find the difference between two functions using function subtraction. In this article, we will answer some frequently asked questions about function subtraction.

Q: What is function subtraction?

A: Function subtraction is a way of combining two functions to create a new function. It is denoted as $(f-g)(x)$, where $f(x)$ and $g(x)$ are the two functions being subtracted.

Q: How do I subtract two functions?

A: To subtract two functions, you need to subtract the corresponding terms in the two functions. For example, if $f(x) = 4x^2 + 1$ and $g(x) = x^2 - 5$, then $(f-g)(x) = (4x^2 + 1) - (x^2 - 5)$.

Q: What is the difference between function subtraction and function addition?

A: Function subtraction and function addition are two different operations. Function subtraction involves finding the difference between two functions, while function addition involves finding the sum of two functions.

Q: Can I subtract a constant from a function?

A: Yes, you can subtract a constant from a function. For example, if $f(x) = 2x^2 + 3$ and you subtract 2 from it, then the new function is $f(x) = 2x^2 + 1$.

Q: Can I subtract a function from a constant?

A: No, you cannot subtract a function from a constant. The result of subtracting a function from a constant is undefined.

Q: What is the order of operations for function subtraction?

A: The order of operations for function subtraction is the same as for regular subtraction. You need to subtract the corresponding terms in the two functions, and then combine like terms.

Q: Can I use function subtraction to find the difference between two polynomials?

A: Yes, you can use function subtraction to find the difference between two polynomials. For example, if $f(x) = 2x^2 + 3x + 1$ and $g(x) = x^2 + 2x - 3$, then $(f-g)(x) = (2x^2 + 3x + 1) - (x^2 + 2x - 3)$.

Q: Can I use function subtraction to find the difference between two rational functions?

A: Yes, you can use function subtraction to find the difference between two rational functions. For example, if $f(x) = \frac{2x^2 + 3x + 1}{x^2 + 1}$ and $g(x) = \frac{x^2 + 2x - 3}{x^2 + 1}$, then $(f-g)(x) = \frac{(2x^2 + 3x + 1) - (x^2 + 2x - 3)}{x^2 + 1}$.

Q: Can I use function subtraction to find the difference between two trigonometric functions?

A: Yes, you can use function subtraction to find the difference between two trigonometric functions. For example, if $f(x) = \sin(x) + \cos(x)$ and $g(x) = \sin(x) - \cos(x)$, then $(f-g)(x) = (\sin(x) + \cos(x)) - (\sin(x) - \cos(x))$.

Conclusion

In this article, we answered some frequently asked questions about function subtraction. We discussed the concept of function subtraction, how to subtract two functions, and the order of operations for function subtraction. We also provided examples of using function subtraction to find the difference between two polynomials, rational functions, and trigonometric functions.