Question

Determine the formula for the inverse of the function\(f(x) = 2 - {e^x}\). Sketch the graph of \(f,{f^{ - 1}}\) and \(y = x\), check whether graphs \(f\) and \({f^{ - 1}}\) reflects about the line \(y = x\).

Short answer

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

Step-by-step solution

Given data

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

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.

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

Then, interchange \(x\) and \(y\), obtain the inverse function, \(y = \ln (2 - x)\) and \(y = x\)on the same axis.

From Figure, it is observed that the graph of \({f^{ - 1}}(x) = \ln (2 - x)\) is a reflection of the graph \(f(x) = 2 - {e^x}\) about the line \(y = x\).