In order to get the inverse of the function `f(x)` which we denote as `f^(-1)(x)`, we simply interchange x and y, and then solve for y:
`y = log_12 x`
Interchanging x and y:
`x = log_12 y`
To solve for y, we note that:
`y = log_a b` is equivalent to `a^y = b` , using definitions of log and exponents.
Hence, we can solve for y` `as:
`y = 12^x`
This means that:
`f^(-1)(x) = 12^x`
To check, note that `f(f^(-1)(x)) = f^(-1)(f(x)) = x`
`f^(-1)(f(x)) = 12^(log_12 x) = x` also,
`f(f^(-1)x)) = log_12 (12^x) = x.`
The inverse function of the function f(x) = log(base 12)x is f^-1(x) = 12^x.