Question

Find a formula for the inverse of the function.

28. \(y = 3\ln \left( {x - 2} \right)\)

Short answer

The inverse function is \(y = 2 + {e^{\frac{x}{3}}}\).

Step-by-step solution

Condition to find the inverse function of a one-to-one function

Step 1: Write the equation as \(y = f\left( x \right)\).

Step 2: If possible, solve the equation in terms of \(y\).

Step 3: Express the inverse \({f^{ - 1}}\) as a function of \(x\), for that interchange\(x\) and \(y\). The equation becomes \(y = {f^{ - 1}}\left( x \right)\).

Determine the formula for the inverse of the function 

Solve the given equation for \(x\) as shown below:

\(\begin{aligned}\frac{y}{3} &= \ln \left( {x - 2} \right)\\{e^{\frac{y}{3}}} &= x - 2\\x &= 2 + {e^{\frac{y}{3}}}\end{aligned}\)

Interchange \(x\) and \(y\) in the above equation as shown below:

\(y = 2 + {e^{\frac{x}{3}}}\)

Thus, the inverse function is \(y = 2 + {e^{\frac{x}{3}}}\).