What is the inverse of the function $f(x)=\frac{1}{4} x-12$?
The problem asks us to find the inverse function \(h(x) = f^{-1}(x)\) of the given linear function \(f(x) = \frac{1}{4}x - 12\). The inverse function essentially reverses the operations performed by the original function, mapping the output back to the input. We denote the inverse as \(h(x)\), such that \(f(h(x)) = x\) and \(h(f(x)) = x\). For any function \(y = f(x)\), the inverse can be found by swapping the roles of \(x\) and \(y\) and solving the resulting equation for \(y\).
To start, we write the original function as an equation:
\[y = \frac{1}{4}x - 12.\]
To find the inverse, we swap \(x\) and \(y\):
\[x = \frac{1}{4}y - 12.\]
Now, we must isolate \(y\) to express it in terms of \(x\). First, add \(12\) to both sides of the equation:
\[x + 12 = \frac{1}{4}y.\]
Next, multiply both sides of the equation by \(4\) to solve for \(y\):
\[4(x + 12) = y.\]
Distributing the \(4\), we get:
\[y = 4x + 48.\]
Therefore, the inverse function is \(h(x) = 4x + 48\).
We compare our result, \(h(x) = 4x + 48\), with the provided options:
Our derived function \(h(x) = 4x + 48\) matches the fourth option exactly. Thus, the inverse of \(f(x) = \frac{1}{4}x - 12\) is \(h(x) = 4x + 48\), and the final answer is \(\boxed{h(x) = 4x + 48}\).
Inverse functions are frequently used in everyday conversions. For instance, if you have a rule for converting temperature from Celsius to Fahrenheit, like \(F = \frac{9}{5}C + 32\), the inverse function allows you to quickly convert Fahrenheit back to Celsius: \(C = \frac{5}{9}(F - 32)\). This logic is applied in fields ranging from programming and data science to currency conversion and manufacturing, where you often need to reverse a process to retrieve original information.