How to Find the Inverse of a Function Like y e^x/(1 - 2e^x)

In this article, we will go through the detailed steps to find the inverse of a function, specifically the function y e^x/(1 - 2e^x). Understanding how to find the inverse of a function is crucial in various mathematical applications, including calculus, algebra, and more. This guide will cover the process step-by-step and explain the necessary calculations clearly.

What is an Inverse Function?

An inverse function is a function that undoes the operation of another function. If we have a function y f(x), its inverse function, denoted as g(y), satisfies the condition that g(f(x)) x and f(g(y)) y. In other words, applying a function and then its inverse (or vice versa) returns the original input.

Steps to Find the Inverse of a Function

To find the inverse of the function y e^x/(1 - 2e^x), we will follow these steps:

Step 1: Express (x) in terms of (y)

Given the equation:

y e^x/(1 - 2e^x)

Step 1.1: Multiply both sides by (1 - 2e^x):
y(1 - 2e^x) e^x

Step 1.2: Expand the equation:
y - 2ye^x e^x

Step 1.3: Rearrange the equation to isolate the term with (e^x):
2ye^x e^x y

Step 1.4: Factor out (e^x):
e^x(2y 1) y

Step 1.5: Solve for (e^x):
e^x y/(2y 1)

Note: It is important to ensure that the solution for (e^x) is mathematically valid, i.e., (2y 1 eq 0).

Step 2: Solve for (x)

Step 2.1: Take the natural logarithm of both sides:
x ln(y/(2y 1))

Note: The natural logarithm is only defined for positive numbers, so (y/(2y 1) > 0).

Step 3: Swap (x) and (y)

Step 3.1: Swap (x) and (y):
y ln(x/(2x 1))

Domain and Range Considerations

To ensure the inverse function is well-defined, we must first define the domain and range of the original function and the inverse function:

Domain of the Original Function: The function (f(x) e^x/(1 - 2e^x)) is defined for all (x) such that (1 - 2e^x eq 0). This implies (e^x eq 1/2), hence (x eq ln(1/2)).

Range of the Inverse Function: The inverse function (f^{-1}(x) ln(x/(2x 1))) is valid for (x) in the range where (x/(2x 1) > 0). Solving this inequality, we find (x

Check for Inverse Function: To verify, we can check the composition of the original function and its inverse. If (f(f^{-1}(x)) x) and (f^{-1}(f(x)) x), then the inverse function is correct.

Conclusion

In conclusion, the inverse of the function (y e^x/(1 - 2e^x)) is given by the function:

f^{-1}(x) ln(x/(2x 1))

Understanding and applying these steps will help in finding the inverse of various functions, which is a fundamental skill in advanced mathematics.