Step-by-step explanation:
We need to show whether
[tex]f^{-1}(x) = g(x)[/tex]
or
[tex]g^{-1}(x) = f(x)[/tex]
so we'll do either one of them,
we'll convert f(x) to f^-1(x) and lets see if it looks like g(x).
[tex]f(x) = e^x - 1[/tex]
we can also write it as:
[tex]y = e^x - 1[/tex]
now all we have to do is to make x the subject of the equation.
[tex]y+1 = e^x[/tex]
[tex]\ln{(y+1)} = x[/tex]
[tex]x=\ln{(y+1)}[/tex]
now we'll interchange the variables
[tex]y=\ln{(x+1)}[/tex]
this is the inverse of f(x)
[tex]f^{-1}(x)=\ln{(x+1)}[/tex]
and it does equal to g(x)
[tex]g(x)=\ln{(x+1)}[/tex]
Hence, both functions are inverse of each other!
This can be shown graphically too:
we can see that both functions are reflections of each other about the line y=x.
