Swap the x and y variables

Oct 13, 2025 by miodeya.com 0 views

**Inverse Functions: Understanding the Concept and Identifying the Correct Inverse**

What is an Inverse Function?

An inverse function is a function that reverses the operation of another function. In other words, if we have a function f(x) that maps an input x to an output f(x), then the inverse function f^(-1)(x) maps the output f(x) back to the input x. Inverse functions are denoted by a superscript "-1" and are used to "undo" the operation of the original function.

Why are Inverse Functions Important?

Inverse functions are important in mathematics because they allow us to solve equations and find the values of unknown variables. For example, if we have an equation f(x) = y, we can use the inverse function f^(-1)(x) to find the value of x that corresponds to a given value of y.

How to Identify the Inverse of a Function

To identify the inverse of a function, we need to follow these steps:

Swap the x and y variables: Swap the x and y variables in the original function. This will give us a new function with x and y variables swapped.
Solve for y: Solve the new function for y. This will give us the inverse function.
Check the domain and range: Check the domain and range of the original function and the inverse function to make sure they are consistent.

Example: Finding the Inverse of y = 3^x

Let's use the function y = 3^x as an example. To find the inverse of this function, we need to follow the steps above.

Swap the x and y variables: Swap the x and y variables in the original function. This gives us x = 3^y.
Solve for y: Solve the new function for y. This gives us y = log_3(x).
Check the domain and range: Check the domain and range of the original function and the inverse function to make sure they are consistent. The original function has a domain of all real numbers and a range of all positive real numbers. The inverse function has a domain of all positive real numbers and a range of all real numbers.

Answer to the Original Question

Based on the steps above, we can see that the inverse of y = 3^x is indeed y = log_3(x). Therefore, the correct answer is:

B. y = log_3 x

Q&A

Q: What is the inverse of the function y = 2x + 1?

A: To find the inverse of this function, we need to follow the steps above. Swapping the x and y variables gives us x = 2y + 1. Solving for y gives us y = (x - 1)/2. Therefore, the inverse of the function y = 2x + 1 is y = (x - 1)/2.

Q: What is the inverse of the function y = x^2?

A: To find the inverse of this function, we need to follow the steps above. Swapping the x and y variables gives us x = y^2. Solving for y gives us y = ±√x. Therefore, the inverse of the function y = x^2 is y = ±√x.

Q: What is the inverse of the function y = 1/x?

A: To find the inverse of this function, we need to follow the steps above. Swapping the x and y variables gives us x = 1/y. Solving for y gives us y = 1/x. Therefore, the inverse of the function y = 1/x is y = 1/x.

Q: What is the inverse of the function y = sin(x)?

A: To find the inverse of this function, we need to follow the steps above. Swapping the x and y variables gives us x = sin(y). Solving for y gives us y = arcsin(x). Therefore, the inverse of the function y = sin(x) is y = arcsin(x).

Conclusion

Inverse functions are an important concept in mathematics that allow us to solve equations and find the values of unknown variables. By following the steps above, we can identify the inverse of a function and use it to solve equations and find the values of unknown variables.