Question

Find a formula for the inverse of the function\(y = \ln (x + 3)\).

Short answer

The formula for the inverse of the function \(y = \ln (x + 3)\) is \({f^{ - 1}}(x) = {e^x} - 3\).

Step-by-step solution

Given data

The given function is \(y = \ln (x + 3)\).

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.

\(\begin{array}{c}y = \ln \left( {x + 3} \right)\\{e^y} = x + 3\\{e^y} - 3 = x\end{array}\)

Take exponential on both the sides.

Now, interchange \(x\) and \(y\), obtain the inverse function, \(y = {e^x} - 3\).

Thus, the required inverse function is \({f^{ - 1}}(x) = {e^x} - 3\).