Answer:
[tex]f^{-1}(x)=\ln{x}[/tex]
Step-by-step explanation:
An inverse function is any function that "undoes" another function. If we think of the function [tex]f[/tex] as some kind of machine that takes in a number [tex]x[/tex] as input and produces a number [tex]f(x)[/tex] as an output, when we give inverse function [tex]f^{-1}[/tex] the number [tex]f(x)[/tex] as in input, we get [tex]x[/tex], our original input, as the output [tex]f^{-1}(x)[/tex]
We need a function that undoes [tex]f(x)=e^x[/tex], and the natural choice for undoing an exponent is with a logarithm. Here, our base is [tex]e[/tex], so we'll choose [tex]\log_e{x}=\ln{x}[/tex] as our inverse function. Let's see how that works:
[tex]f^{-1}(f(x))=\ln{e^x}[/tex]
[tex]\ln{e^x}[/tex] is the power we have to raise [tex]e[/tex] to to get [tex]e^x[/tex], which is [tex]x[/tex], so
[tex]f^{-1}(f(x))=\ln{e^x}=x[/tex]
And we have our function.
