If [tex]f(x)=9 x[/tex], which of the following is the inverse of [tex]f(x)[/tex]?
A. [tex]f^{-1}(x)=x+9[/tex]
B. [tex]f^{-1}(x)=x-9[/tex]
C. [tex]f^{-1}(x)=\frac{x}{9}[/tex]
D. [tex]f^{-1}(x)=\frac{9}{x}[/tex]
Answer (2)
Replace f ( x ) with y : y = 9 x .
Swap x and y : x = 9 y .
Solve for y : y = 9 x .
Replace y with f − 1 ( x ) : f − 1 ( x ) = 9 x . The answer is 9 x .
Explanation
Understanding the Problem We are given the function f ( x ) = 9 x and we want to find its inverse, denoted as f − 1 ( x ) . The inverse function essentially 'undoes' what the original function does.
Finding the Inverse Function To find the inverse, we can follow these steps:
Replace f ( x ) with y : y = 9 x .
Swap x and y : x = 9 y .
Solve for y in terms of x : To isolate y , we divide both sides of the equation by 9:
9 x = 9 9 y
9 x = y
Replace y with f − 1 ( x ) : f − 1 ( x ) = 9 x .
Determining the Answer Therefore, the inverse of the function f ( x ) = 9 x is f − 1 ( x ) = 9 x . Comparing this to the given options, we see that it matches option C.
Examples
Imagine you're converting meters to centimeters. If f ( x ) = 100 x converts meters ( x ) to centimeters ( f ( x ) ), then the inverse function f − 1 ( x ) = 100 x converts centimeters back to meters. This concept is useful in unit conversions, decoding messages, or reversing any process where you know the original function.
The inverse of the function f ( x ) = 9 x is f − 1 ( x ) = 9 x . Therefore, the correct answer is option C. This means that if you input a number into the inverse function, you will get back the original number that was multiplied by 9 in the equation.
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