Question

Determine the value of \({g^{ - 1}}(4)\) if \(g(x) = 3 + x + {e^x}\).

Short answer

The value of \({g^{ - 1}}(4) = 0\).

Step-by-step solution

Given data

The given function is \(g(x) = 3 + x + {e^x}\).

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 \({\rm{X}}\).

Simplify the expression

The given function is \(g(x) = 3 + x + {e^x}\).

Notice that the value of \(g(0) = 4\) as, \(3 + 0 + {e^4} = 4\).

According to the definition of an inverse function, \({f^{ - 1}}(y) = x\) if and only if \(f(x) = y\) where \(f\) is one-to-one.

So, it can be concluded that \({g^{ - 1}}(4) = 0\) as \(g(0) = 4\).

Therefore, the value of \({g^{ - 1}}(4) = 0\).