Which function is the inverse of $f(x)=9 x-3$?
The objective is to find the inverse of the linear function \(f(x) = 9x - 3\). An inverse function, denoted by \(f^{-1}(x)\), essentially reverses the operations of the original function. If the function \(f\) maps an input \(x\) to an output \(y\), such that \(f(x) = y\), then the inverse function maps the output \(y\) back to the original input \(x\), such that \(f^{-1}(y) = x\). We seek the expression for \(f^{-1}(x)\) by finding \(x\) in terms of \(y\) and then swapping the variables.
To find the inverse, we start by setting the function expression equal to \(y\):
\[y = 9x - 3\]
Now, we solve for \(x\) in terms of \(y\). First, add 3 to both sides of the equation to isolate the term with \(x\):
\[y + 3 = 9x\]
Next, divide both sides by 9:
\[x = \frac{y + 3}{9}\]
This equation shows that for any output \(y\), we can retrieve the original input \(x\) using the expression \(\frac{y + 3}{9}\).
Finally, we express the inverse function by replacing the variable \(y\) with \(x\), which gives us the function \(f^{-1}(x)\). By substituting \(x\) back into the expression we found in the previous step:
\[f^{-1}(x) = \frac{x + 3}{9}\]
This is the inverse function of \(f(x) = 9x - 3\). Comparing this result with the provided options, we can see that it matches the second option.
Inverse functions are essential in real-life conversions. For example, if you are traveling between countries with different currencies, a function \(f(x)\) might represent the amount of foreign currency you receive for \(x\) dollars. To find out how many dollars you started with based on the amount of foreign currency you now hold, you would use the inverse function \(f^{-1}(y)\). This is exactly how currency exchange kiosks work internally, reversing the conversion rate to calculate your original amount.