If [tex]f(x)=\frac{1}{9} x-2[/tex], what is [tex]f^{-1}(x)[/tex]?
Answer (1)
Replace f ( x ) with y : y = 9 1 x − 2 .
Swap x and y : x = 9 1 y − 2 .
Solve for y : y = 9 x + 18 .
Replace y with f − 1 ( x ) : f − 1 ( x ) = 9 x + 18 . The inverse function is f − 1 ( x ) = 9 x + 18 .
Explanation
Understanding the Problem We are given the function f ( x ) = 9 1 x − 2 and we want to find its inverse, f − 1 ( x ) . The inverse function essentially reverses the operation of the original function.
Replace f(x) with y To find the inverse function, we first replace f ( x ) with y :
y = 9 1 x − 2
Swap x and y Next, we swap x and y :
x = 9 1 y − 2
Isolate the term with y Now, we solve for y in terms of x . First, we add 2 to both sides of the equation: x + 2 = 9 1 y
Solve for y Then, we multiply both sides by 9 to isolate y :
9 ( x + 2 ) = y
Simplify Distribute the 9: 9 x + 18 = y
Write the inverse function Finally, we replace y with f − 1 ( x ) :
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. In general, inverse functions are useful in any situation where you need to reverse a process or calculation. For example, if a store marks up prices by a certain percentage, the inverse function could be used to calculate the original price before the markup. This concept is fundamental in many areas of science, engineering, and economics where reversing operations is a common task.
