What is the inverse of the function $f(x)=\frac{1}{9} x+2$ ?
We are given the linear function \(f(x) = \frac{1}{9}x + 2\). Our objective is to find its inverse function, denoted as \(h(x)\). By definition, an inverse function \(h(x)\) reverses the operation of the original function \(f(x)\). If \(f(x) = y\), then the inverse must satisfy \(h(y) = x\). Geometrically, this corresponds to reflecting the graph of \(f(x)\) across the line \(y = x\).
To find the inverse function, we start by setting the original function equal to a variable \(y\):
\[y = \frac{1}{9}x + 2.\]
Next, to find the inverse, we swap the roles of \(x\) and \(y\), representing the reflection across the line \(y = x\):
\[x = \frac{1}{9}y + 2.\]
Now, we solve this new equation for \(y\) to express it in terms of \(x\):
First, subtract \(2\) from both sides of the equation:
\[x - 2 = \frac{1}{9}y.\]
Next, multiply both sides of the equation by \(9\) to isolate \(y\):
\[9(x - 2) = y.\]
Distribute the \(9\) across the terms inside the parentheses:
\[y = 9x - 18.\]
Thus, the inverse function is \(h(x) = 9x - 18\).
Comparing our derived result, \(h(x) = 9x - 18\), with the given options:
Our result matches the second option. The inverse of the function \(f(x) = \frac{1}{9}x + 2\) is \(\boxed{h(x) = 9x - 18}\).
Inverse functions are frequently used in everyday situations, such as currency conversion or unit conversions. If a function calculates the price of an item in US Dollars based on its price in another currency, the inverse function allows you to calculate the price in that other currency given a US Dollar amount. For instance, if you have a rule to convert temperature from Celsius to Fahrenheit, the inverse function lets you convert the result back to Celsius, helping you understand the relationship between different units of measure.