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)
The inverse function of f ( x ) = 9 1 x − 2 is f − 1 ( x ) = 9 x + 18 . This means that if you apply the inverse function to the output of the original function, you'll obtain the original input. The correct option is A.
;
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 .
The inverse function is 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 function, 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 :
Starting with x = f r a c 1 9 y − 2 , we want to isolate y .
First, add 2 to both sides of the equation:
x + 2 = f r a c 1 9 y
Next, multiply both sides by 9 to get y by itself:
9 ( x + 2 ) = y
Distribute the 9:
9 x + 18 = y
The Inverse Function Now, we replace y with f − 1 ( x ) :
f − 1 ( x ) = 9 x + 18
Final Answer Therefore, the inverse function is f − 1 ( x ) = 9 x + 18 .
Examples
In real life, inverse functions can be used to convert between different units of measurement. For example, if f ( x ) converts Celsius to Fahrenheit, then f − 1 ( x ) would convert Fahrenheit back to Celsius. Understanding inverse functions helps in reversing processes or converting back to original values after a transformation.
