Find the inverse function of $f(x)=3^{x+2}$.
We are given the exponential function \(f(x) = 3^{x+2}\). Our goal is to find the inverse function \(f^{-1}(x)\). An inverse function essentially reverses the effect of the original function. If the original function maps an input \(x\) to an output \(y\), such that \(f(x) = y\), the inverse function must map \(y\) back to \(x\), i.e., \(f^{-1}(y) = x\). By standard convention, we represent the inverse function using the independent variable \(x\).
To find the inverse, we start by setting the function equal to \(y\):
\[y = 3^{x+2}\]
Next, we interchange the roles of \(x\) and \(y\) to solve for the inverse function:
\[x = 3^{y+2}\]
This equation represents the relationship between the output and the input of the original function. We now need to isolate \(y\) to express it in terms of \(x\).
To isolate \(y\), we apply the logarithm with base 3 to both sides of the equation:
\[\log_3(x) = \log_3(3^{y+2})\]
Using the fundamental property of logarithms, \(\log_b(b^a) = a\), we can simplify the right side of the equation:
\[\log_3(x) = y + 2\]
Finally, to solve for \(y\), we subtract 2 from both sides:
\[y = \log_3(x) - 2\]
Thus, the inverse function is \(f^{-1}(x) = \log_3(x) - 2\).
Inverse functions are crucial in fields like data science and finance, where we often need to "reverse" an operation to retrieve an original input. For example, if you know that the value of an investment grows exponentially over time according to a function \(f(t) = P(1+r)^t\), the inverse function \(f^{-1}(A)\) allows you to calculate the time \(t\) needed to reach a specific target amount \(A\). By applying logarithms, as shown in this problem, you can solve for the time variable accurately.