Select The Correct Answer.Which Function Defines \[$(f-g)(x)\$\]?$\[ \begin{align*} f(x) &= \sqrt{\frac{x}{3}} + 11 \\ g(x) &= 5 + \frac{2}{x} \end{align*} \\]A. \[$(f-g)(x) = \sqrt{\frac{g}{8} - \frac{2}{x}} + 16\$\]B.
Introduction
In mathematics, functions are used to describe relationships between variables. When dealing with functions, it's essential to understand how to perform operations on them, such as addition, subtraction, multiplication, and division. In this article, we will focus on function subtraction, specifically on the function that defines {(f-g)(x)$}$. We will use the given functions {f(x) = \sqrt{\frac{x}{3}} + 11$}$ and {g(x) = 5 + \frac{2}{x}$}$ to illustrate the concept.
Function Subtraction
Function subtraction is a process of subtracting one function from another. It involves taking the difference between the two functions, which is denoted by the symbol {(f-g)(x)$}$. To find the function that defines {(f-g)(x)$}$, we need to subtract the function {g(x)$}$ from the function {f(x)$}$.
Step-by-Step Solution
To find the function that defines {(f-g)(x)$}$, we will follow these steps:
- Write down the given functions: We are given two functions, {f(x) = \sqrt{\frac{x}{3}} + 11$}$ and {g(x) = 5 + \frac{2}{x}$}$.
- Subtract the functions: To find the function that defines {(f-g)(x)$}$, we need to subtract the function {g(x)$}$ from the function {f(x)$}$. This can be done by subtracting the corresponding terms of the two functions.
Calculating the Difference
Let's calculate the difference between the two functions:
{(f-g)(x) = f(x) - g(x)$}$
{(f-g)(x) = \sqrt{\frac{x}{3}} + 11 - (5 + \frac{2}{x})$}$
To simplify the expression, we need to combine like terms:
{(f-g)(x) = \sqrt{\frac{x}{3}} + 11 - 5 - \frac{2}{x}$}$
{(f-g)(x) = \sqrt{\frac{x}{3}} + 6 - \frac{2}{x}$}$
Simplifying the Expression
To simplify the expression further, we can combine the terms under the square root:
{(f-g)(x) = \sqrt{\frac{x}{3}} - \frac{2}{x} + 6$}$
However, we can simplify the expression even further by combining the terms under the square root:
{(f-g)(x) = \sqrt{\frac{x}{3} - \frac{6}{x}} + 6$}$
Final Answer
Based on the calculations above, the function that defines {(f-g)(x)$}$ is:
{(f-g)(x) = \sqrt{\frac{x}{3} - \frac{6}{x}} + 6$}$
This is the correct answer.
Conclusion
In this article, we have learned how to find the function that defines {(f-g)(x)$}$ by subtracting one function from another. We have used the given functions {f(x) = \sqrt{\frac{x}{3}} + 11$}$ and {g(x) = 5 + \frac{2}{x}$}$ to illustrate the concept. We have followed the steps to find the function that defines {(f-g)(x)$}$ and have arrived at the final answer.
References
- [1] Calculus, 3rd edition, Michael Spivak
- [2] Calculus, 2nd edition, James Stewart
Discussion
What do you think about function subtraction? Have you ever encountered a situation where you needed to subtract one function from another? Share your thoughts and experiences in the comments below.
Related Topics
- Function addition
- Function multiplication
- Function division
- Inverse functions
Frequently Asked Questions
- What is function subtraction?
- Function subtraction is a process of subtracting one function from another.
- How do I find the function that defines {(f-g)(x)$}$?
- To find the function that defines {(f-g)(x)$}$, you need to subtract the function {g(x)$}$ from the function {f(x)$}$.
- What is the final answer?
- The final answer is {(f-g)(x) = \sqrt{\frac{x}{3} - \frac{6}{x}} + 6$}$.
Q&A: Function Subtraction ==========================
- The final answer is {(f-g)(x) = \sqrt{\frac{x}{3} - \frac{6}{x}} + 6$}$.
Introduction
In our previous article, we discussed function subtraction and how to find the function that defines {(f-g)(x)$}$. In this article, we will answer some frequently asked questions about function subtraction.
Q: What is function subtraction?
A: Function subtraction is a process of subtracting one function from another. It involves taking the difference between the two functions, which is denoted by the symbol {(f-g)(x)$}$.
Q: How do I find the function that defines {(f-g)(x)$}$?
A: To find the function that defines {(f-g)(x)$}$, you need to subtract the function {g(x)$}$ from the function {f(x)$}$. This can be done by subtracting the corresponding terms of the two functions.
Q: What is the difference between function subtraction and function addition?
A: Function subtraction involves taking the difference between two functions, while function addition involves taking the sum of two functions. For example, if we have two functions {f(x) = x^2 + 1$}$ and {g(x) = x + 1$}$, the function that defines {(f-g)(x)$}$ would be {(f-g)(x) = x^2 + 1 - (x + 1) = x^2 - x$}$, while the function that defines {(f+g)(x)$}$ would be {(f+g)(x) = x^2 + 1 + (x + 1) = x^2 + x + 2$}$.
Q: Can I subtract a constant from a function?
A: Yes, you can subtract a constant from a function. For example, if we have a function {f(x) = x^2 + 1$}$ and we want to subtract 2 from it, the resulting function would be {f(x) - 2 = x^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 would be a constant, not a function.
Q: What is the final answer to the problem {(f-g)(x) = \sqrt{\frac{x}{3} - \frac{6}{x}} + 6$}$?
A: The final answer to the problem {(f-g)(x) = \sqrt{\frac{x}{3} - \frac{6}{x}} + 6$}$ is indeed {(f-g)(x) = \sqrt{\frac{x}{3} - \frac{6}{x}} + 6$}$.
Q: Can I use function subtraction to solve real-world problems?
A: Yes, you can use function subtraction to solve real-world problems. For example, if we have a function that represents the cost of producing a certain product, and we want to find the cost of producing a certain quantity of that product, we can use function subtraction to find the difference between the two functions.
Conclusion
In this article, we have answered some frequently asked questions about function subtraction. We have discussed the concept of function subtraction, how to find the function that defines {(f-g)(x)$}$, and how to use function subtraction to solve real-world problems.
References
- [1] Calculus, 3rd edition, Michael Spivak
- [2] Calculus, 2nd edition, James Stewart
Discussion
What do you think about function subtraction? Have you ever encountered a situation where you needed to subtract one function from another? Share your thoughts and experiences in the comments below.
Related Topics
- Function addition
- Function multiplication
- Function division
- Inverse functions
Frequently Asked Questions
- What is function subtraction?
- Function subtraction is a process of subtracting one function from another.
- How do I find the function that defines {(f-g)(x)$}$?
- To find the function that defines {(f-g)(x)$}$, you need to subtract the function {g(x)$}$ from the function {f(x)$}$.
- What is the difference between function subtraction and function addition?
- Function subtraction involves taking the difference between two functions, while function addition involves taking the sum of two functions.
- Can I subtract a constant from a function?
- Yes, you can subtract a constant from a function.
- Can I subtract a function from a constant?
- No, you cannot subtract a function from a constant.