What is the inverse of the function $f(x)=\frac{1}{9} x+2$ ?
To find the inverse of the linear function \(f(x) = \frac{1}{9}x + 2\), we first define the function as \(y = \frac{1}{9}x + 2\). The inverse function \(f^{-1}(x)\), which we denote as \(h(x)\), exists because linear functions with a non-zero slope are bijective. Our goal is to determine \(h(x)\) by finding a function that reverses the effect of \(f(x)\).
To derive the inverse, we start with the equation \(y = \frac{1}{9}x + 2\). We swap the variables \(x\) and \(y\) to reflect the inverse operation:
\[x = \frac{1}{9}y + 2\]
Now, we solve this new equation for \(y\). First, subtract 2 from both sides of the equation:
\[x - 2 = \frac{1}{9}y\]
Next, multiply both sides by 9 to isolate \(y\):
\[9(x - 2) = y\]
\[y = 9x - 18\]
Having solved for \(y\), we have found the inverse function \(h(x) = 9x - 18\). This can be verified by checking that \(f(h(x)) = x\) and \(h(f(x)) = x\).
Substituting \(h(x)\) into \(f(x)\):
\[f(h(x)) = \frac{1}{9}(9x - 18) + 2 = (x - 2) + 2 = x\]
This confirms our result. Comparing this to the given options, the correct inverse is \(h(x) = 9x - 18\).
Imagine a machine that takes an input number, divides it by 9, and then adds 2 to get an output. If you want to reverse this process to recover the original number from the output, you would need to perform the inverse operations in the opposite order. First, you would subtract 2 from the output, and then multiply the result by 9. This concept of reversing operations is the core of finding the inverse of a function, which is essential for solving equations and understanding relationships between variables in many scientific and everyday fields.