If [tex]$f(x)=\frac{1}{9} x-2$[/tex], what is [tex]$f^{-1}(x)$[/tex]?
A. [tex]$f^{-1}(x)=9 x+18$[/tex]
B. [tex]$f^{-1}(x)=\frac{1}{9} x+2$[/tex]
C. [tex]$f^{-1}(x)=9 x+2$[/tex]
D. [tex]$f^{-1}(x)=-2 x+\frac{1}{9}$[/tex]
Answer (2)
Replace f ( x ) with y : y = f r a c 1 9 x − 2 .
Swap x and y : x = f r a c 1 9 y − 2 .
Solve for y : y = 9 x + 18 .
Replace y with f − 1 ( x ) : f − 1 ( x ) = 9 x + 18 .
Explanation
Understanding the Problem We are given the function f ( x ) = f r a c 1 9 x − 2 and we want to find its inverse, f − 1 ( x ) . The inverse function essentially 'undoes' what the original function does.
Finding the Inverse To find the inverse, we can follow these steps:
Replace f ( x ) with y : y = f r a c 1 9 x − 2 .
Swap x and y : x = f r a c 1 9 y − 2 .
Solve for y in terms of x .
Solving for y Let's solve for y :
Add 2 to both sides of the equation: x + 2 = f r a c 1 9 y .
Multiply both sides by 9 to isolate y : 9 ( x + 2 ) = y .
Distribute the 9: 9 x + 18 = y .
Writing the Inverse Function Now, replace y with f − 1 ( x ) to denote the inverse function: f − 1 ( x ) = 9 x + 18 .
Final Answer Therefore, the inverse function is f − 1 ( x ) = 9 x + 18 .
Examples
Imagine you're converting temperatures from Celsius to Fahrenheit using a function. The inverse function would then convert Fahrenheit back to Celsius. Similarly, if you have a function that encodes a message, the inverse function would decode it. Inverse functions are useful in any situation where you need to reverse a process or calculation.
The inverse function of f ( x ) = 9 1 x − 2 is f − 1 ( x ) = 9 x + 18 . Therefore, the correct answer is option A. This means for every output you get from the function, the inverse will return you to the original input value.
;
