Question

(a) Find the domain of \(f\left( x \right) = \ln \left( {{e^x} - 3} \right)\)

(b) Find \({f^{ - 1}}\) and its domain.

Short answer

(a) The domain of the function is \(\left( {\ln 3,\infty } \right)\).

(b) The domain of the inverse function \({f^{ - 1}}\) is \(\mathbb{R}\).

Step-by-step solution

Determine the domain of the function

a)

The function \(f\left( x \right) = \ln \left( {{e^x} - 3} \right)\) is defined when \({e^x} - 3 > 0\).

The domain of the function is shown below:

\(\begin{aligned}{e^x} - 3 > 0\\{e^x} > 3\\x > \ln 3\end{aligned}\)

Thus, the domain of \(f\left( x \right) = \ln \left( {{e^x} - 3} \right)\) is \(\left( {\ln 3,\infty } \right)\).

Determine the inverse function \({f^{ - 1}}\) and its domain

b)

Write the equation as \(y = f\left( x \right) = \ln \left( {{e^x} - 3} \right)\) and solve the equation for \(x\) as shown below:

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

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

\(y = \ln \left( {{e^x} + 3} \right)\)

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

The domain of the inverse function is shown below:

\(\begin{aligned}{e^x} + 3 > 0\\{e^x} > - 3\end{aligned}\)

It is true for each real \(x\).

Thus, the domain of \({f^{ - 1}}\) is the set of all real numbers or \(\mathbb{R}\).