Question

If \(g\left( x \right) = {\bf{3}} + x + {e^x},\) find \({g^{ - 1}}\left( {\bf{4}} \right)\).

Short answer

The inverse function is \({g^{ - 1}}\left( 4 \right) = 0\).

Step-by-step solution

Condition for one-to-one function

If a function never takes on the same value again, it is called a one-to-one function. This means that;

\(f\left( {{x_1}} \right) \ne f\left( {{x_2}} \right)\) whenever \({x_1} \ne {x_2}\)

Determine \({g^{ - 1}}\left( 4 \right)\)

To begin, we must find \(x\) such that \(g\left( x \right) = 4\).

According to the inspection, we can see that if \(x = 0\), then \(g\left( x \right) = 4\). \({g^{ - 1}}\left( 4 \right) = 0\) because \(f\) is a one-to-one function (and an increasing function), it has an inverse.

Thus, \({g^{ - 1}}\left( 4 \right) = 0\).