Question

Find a formula for the inverse of the function\(f(x) = {e^{2x - 1}}\).

Short answer

The formula for the inverse of the function \(f(x) = {e^{2x - 1}}\) is \({f^{ - 1}}(x) = \frac{{\ln x + 1}}{2}\).

Step-by-step solution

Given data

The given function is \(y = {e^{2x - 1}}\).

Concept of functions

The simplest definition is an equation will be a function if, for any \({\rm{x}}\) in the domain of the equation (the domain is all the \({\rm{x}}\)'s that can be plugged into the equation), the equation will yield exactly one value of \({\rm{y}}\) when we evaluate the equation at a specific \({\rm{X}}\).

Solve the equation

Solve this equation for \(x\) as shown below.

\(\begin{array}{c}y = \ln {e^{2x - 1}}\\\ln y = 2x - 1\\\ln y + 1 = 2x\\x = \frac{{\ln y + 1}}{2}\end{array}\)

Then, interchange \(x\) and \(y\), obtain the inverse function, \(y = \frac{{\ln x + 1}}{2}\).

Thus, the required inverse function is \({f^{ - 1}}(x) = \frac{{\ln x + 1}}{2}\).