Comparison with Possible Answers

Oct 13, 2025 by windailys.com 51 views

Introduction

In mathematics, the concept of inverse functions is crucial in understanding the relationship between two functions. Given a function f(x), the inverse function f^(-1)(x) is a function that undoes the action of f(x). In other words, if f(x) maps an input x to an output y, then f^(-1)(x) maps the output y back to the input x. In this article, we will focus on finding the inverse function of a given function f(x) = (x^(1/5))/3 + 10.

Step 1: Understand the Given Function

The given function is f(x) = (x^(1/5))/3 + 10. This function takes an input x and maps it to an output y. To find the inverse function, we need to understand the behavior of this function. The function has a fractional exponent, which indicates that it is a power function. The power function is a type of function that has the form f(x) = x^n, where n is a constant.

Step 2: Replace f(x) with y

To find the inverse function, we need to replace f(x) with y. This means that we will rewrite the function as y = (x^(1/5))/3 + 10.

Step 3: Interchange x and y

Next, we need to interchange x and y. This means that we will rewrite the function as x = (y^(1/5))/3 + 10.

Step 4: Solve for y

Now, we need to solve for y. To do this, we will isolate y on one side of the equation. We can start by subtracting 10 from both sides of the equation: x - 10 = (y^(1/5))/3.

Step 5: Multiply both sides by 3

Next, we will multiply both sides of the equation by 3 to get rid of the fraction: 3(x - 10) = y^(1/5).

Step 6: Raise both sides to the power of 5

Now, we will raise both sides of the equation to the power of 5 to get rid of the fractional exponent: (3(x - 10))^5 = y.

Step 7: Simplify the equation

Finally, we can simplify the equation by removing the parentheses: 3^5(x - 10)^5 = y.

Step 8: Write the inverse function

Now that we have solved for y, we can write the inverse function as f^(-1)(x) = 3^5(x - 10)^5.

Conclusion

In this article, we have found the inverse function of the given function f(x) = (x^(1/5))/3 + 10. The inverse function is f^(-1)(x) = 3^5(x - 10)^5. This function takes an input x and maps it to an output y, which is the original input x.

Comparison with Possible Answers

Now, let's compare our answer with the possible answers given:

A. f^(-1)(x) = 3(x - 10)^5 B. f^(-1)(x) = 3(x - 1) C. f^(-1)(x) = 3(x^5 - 10)

Our answer is different from all the possible answers. However, we can see that our answer is similar to answer A, but with a different exponent.

Discussion

The concept of inverse functions is crucial in mathematics, and it has many applications in various fields. In this article, we have found the inverse function of a given function f(x) = (x^(1/5))/3 + 10. The inverse function is f^(-1)(x) = 3^5(x - 10)^5. This function takes an input x and maps it to an output y, which is the original input x.

Final Answer

Q: What is the concept of inverse functions?

A: In mathematics, the concept of inverse functions is crucial in understanding the relationship between two functions. Given a function f(x), the inverse function f^(-1)(x) is a function that undoes the action of f(x). In other words, if f(x) maps an input x to an output y, then f^(-1)(x) maps the output y back to the input x.

Q: How do I find the inverse function of a given function?

A: To find the inverse function of a given function f(x), you need to follow these steps:

Replace f(x) with y.
Interchange x and y.
Solve for y.
Raise both sides to the power of 5 (if the original function has a fractional exponent).
Simplify the equation.

Q: What if the original function has a fractional exponent?

A: If the original function has a fractional exponent, you need to raise both sides of the equation to the power of 5 to get rid of the fractional exponent.

Q: How do I know if the inverse function is correct?

A: To check if the inverse function is correct, you can plug in the original function and the inverse function into each other and see if you get the original input x.

Q: What if the inverse function is not one-to-one?

A: If the inverse function is not one-to-one, it means that the original function is not invertible. In this case, you need to find a different function that is invertible.

Q: Can I use the inverse function to solve equations?

A: Yes, you can use the inverse function to solve equations. For example, if you have an equation f(x) = y, you can use the inverse function f^(-1)(x) to solve for x.

Q: How do I find the inverse function of a composite function?

A: To find the inverse function of a composite function, you need to follow these steps:

Find the inverse function of each individual function in the composite function.
Use the inverse functions to create a new function that undoes the action of the composite function.

Q: What if the composite function has multiple functions?

A: If the composite function has multiple functions, you need to find the inverse function of each individual function and then use the inverse functions to create a new function that undoes the action of the composite function.

Q: Can I use the inverse function to solve optimization problems?

A: Yes, you can use the inverse function to solve optimization problems. For example, if you have a function f(x) that you want to maximize or minimize, you can use the inverse function f^(-1)(x) to find the maximum or minimum value of the function.

Q: How do I find the inverse function of a function with multiple variables?

A: To find the inverse function of a function with multiple variables, you need to follow these steps:

Replace the function with y.
Interchange x and y.
Solve for y.
Use the inverse functions to create a new function that undoes the action of the original function.

Q: What if the function has multiple variables and multiple functions?

A: If the function has multiple variables and multiple functions, you need to find the inverse function of each individual function and then use the inverse functions to create a new function that undoes the action of the original function.

Conclusion

In this article, we have answered some common questions about finding the inverse function of a given function. We have also provided some tips and tricks for finding the inverse function of composite functions and functions with multiple variables. We hope that this article has been helpful in understanding the concept of inverse functions and how to find them.