Select The Correct Answer.Which Function Defines \[$(g-f)(x)\$\]?$\[ \begin{array}{l} f(x) = \sqrt[3]{12x+1} + 4 \\ g(x) = \log(x-3) + 6 \end{array} \\]A. \[$(g-f)(x) = \log(x-3) - \sqrt[3]{12x+1} + 10\$\]B. \[$(g-f)(x) =

Introduction to Function Subtraction

When dealing with functions, subtraction is a crucial operation that allows us to find the difference between two functions. In this article, we will explore the concept of subtracting functions and how to define the correct function that represents the difference between two given functions.

Understanding the Given Functions

To begin with, let's examine the two given functions:

Function f(x)

The function f(x) is defined as:

$$f(x) = \sqrt[3]{12x+1} + 4$$

This function involves a cube root and a constant term.

Function g(x)

The function g(x) is defined as:

$$g(x) = \log(x-3) + 6$$

This function involves a logarithmic term and a constant term.

Subtracting Functions: The Concept

When subtracting functions, we need to subtract the value of one function from the value of another function. In this case, we want to find the function that represents the difference between g(x) and f(x).

Defining the Correct Function

To define the correct function that represents the difference between g(x) and f(x), we need to subtract f(x) from g(x). This can be represented as:

$$(g-f)(x) = g(x) - f(x)$$

Calculating the Difference

Now, let's calculate the difference between g(x) and f(x):

$$(g-f)(x) = g(x) - f(x)$$

$$\begin{aligned} (g-f)(x) &= \log(x-3) + 6 - (\sqrt[3]{12x+1} + 4) \\ &= \log(x-3) + 6 - \sqrt[3]{12x+1} - 4 \\ &= \log(x-3) - \sqrt[3]{12x+1} + 2 \end{aligned}$$

Conclusion

In conclusion, the correct function that defines $(g-f)(x)$ is:

$$(g-f)(x) = \log(x-3) - \sqrt[3]{12x+1} + 2$$

This function represents the difference between g(x) and f(x).

Answer Selection

Based on our calculation, the correct answer is:

A. $(g-f)(x) = \log(x-3) - \sqrt[3]{12x+1} + 2$

This answer matches the function we derived in the previous section.

Final Thoughts

In this article, we explored the concept of subtracting functions and how to define the correct function that represents the difference between two given functions. We examined the given functions f(x) and g(x) and calculated the difference between them. The correct function that defines $(g-f)(x)$ is:

$$(g-f)(x) = \log(x-3) - \sqrt[3]{12x+1} + 2$$

This function represents the difference between g(x) and f(x).

Introduction

In our previous article, we explored the concept of subtracting functions and how to define the correct function that represents the difference between two given functions. In this article, we will provide a Q&A guide to help you better understand the concept of subtracting functions.

Q1: What is the difference between subtracting functions and adding functions?

A1: Subtracting functions involves finding the difference between two functions, whereas adding functions involves finding the sum of two functions. When subtracting functions, we need to subtract the value of one function from the value of another function.

Q2: How do I subtract two functions?

A2: To subtract two functions, we need to follow the order of operations (PEMDAS):

1. Evaluate the expressions inside the parentheses.
2. Exponentiate any exponential expressions.
3. Multiply and divide any expressions from left to right.
4. Add and subtract any expressions from left to right.

Q3: What is the correct order of operations when subtracting functions?

A3: The correct order of operations when subtracting functions is:

1. Subtract the constant terms.
2. Subtract the coefficients of the variables.
3. Subtract the variables.

Q4: How do I handle negative exponents when subtracting functions?

A4: When subtracting functions with negative exponents, we need to follow the rule:

$$\frac{1}{x^n} = x^{-n}$$

Q5: Can I subtract functions with different variables?

A5: Yes, you can subtract functions with different variables. However, you need to make sure that the variables are defined and that the functions are well-defined.

Q6: How do I handle functions with absolute values when subtracting functions?

A6: When subtracting functions with absolute values, we need to follow the rule:

$$|a-b| = |b-a|$$

Q7: Can I subtract functions with different domains?

A7: Yes, you can subtract functions with different domains. However, you need to make sure that the domains are compatible and that the functions are well-defined.

Q8: How do I handle functions with complex numbers when subtracting functions?

A8: When subtracting functions with complex numbers, we need to follow the rules of complex arithmetic:

1. Add and subtract complex numbers by adding and subtracting their real and imaginary parts separately.
2. Multiply and divide complex numbers by multiplying and dividing their magnitudes and angles separately.

Q9: Can I subtract functions with different ranges?

A9: Yes, you can subtract functions with different ranges. However, you need to make sure that the ranges are compatible and that the functions are well-defined.

Q10: How do I handle functions with trigonometric functions when subtracting functions?

A10: When subtracting functions with trigonometric functions, we need to follow the rules of trigonometric arithmetic:

1. Add and subtract trigonometric functions by adding and subtracting their values separately.
2. Multiply and divide trigonometric functions by multiplying and dividing their values separately.

Conclusion

In this article, we provided a Q&A guide to help you better understand the concept of subtracting functions. We covered various topics, including the difference between subtracting and adding functions, the correct order of operations, and how to handle different types of functions. We hope this guide has been helpful in your understanding of subtracting functions.