Find the inverse of the given function:
$f(x)=4 x-12$
The function provided is \(f(x) = 4x - 12\). This is a linear function of the form \(f(x) = ax + b\), where \(a = 4\) and \(b = -12\). Because the slope \(a = 4\) is non-zero, the function is a bijection (it is both one-to-one and onto), which confirms that it possesses a unique inverse function, denoted as \(f^{-1}(x)\). The goal is to find this inverse function.
To find the inverse function, we first express the function as an equation in terms of \(x\) and \(y\):
\[y = 4x - 12.\]
Next, we solve for \(x\) in terms of \(y\). We begin by adding \(12\) to both sides of the equation:
\[y + 12 = 4x.\]
Then, we divide both sides by \(4\):
\[x = \frac{y + 12}{4}.\]
This expression gives us the original input \(x\) as a function of the output \(y\).
To express the inverse as a function of \(x\), we swap the variables \(x\) and \(y\). Thus, the inverse function is:
\[f^{-1}(x) = \frac{x + 12}{4}.\]
We can also write this as:
\[f^{-1}(x) = \frac{1}{4}x + 3.\]
To verify our result, we check if \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\).
First, \(f(f^{-1}(x)) = 4\left(\frac{x+12}{4}\right) - 12 = (x+12) - 12 = x\).
Second, \(f^{-1}(f(x)) = \frac{(4x-12)+12}{4} = \frac{4x}{4} = x\).
Both conditions are satisfied, confirming that our inverse function is correct. Therefore, the inverse of \(f(x) = 4x - 12\) is \(f^{-1}(x) = \frac{x+12}{4}\).
The concept of inverse functions is used in real-life whenever we need to reverse a process. For example, if a company has a pricing model where the final price \(y\) is calculated based on an item's wholesale cost \(x\) using a formula like \(f(x) = 4x - 12\) (perhaps representing a fixed markup and discount), a manager can use the inverse function \(f^{-1}(y) = \frac{y+12}{4}\) to quickly determine what the original wholesale cost \(x\) was, given only the final retail price \(y\). This is essential for accounting, supply chain management, and financial analysis.