If [tex]f(x)=9 x-8[/tex], which of the following is the inverse of [tex]f(x)[/tex]?
A. [tex]f^{-1}(x)=\frac{x+9}{8}[/tex]
B. [tex]f^{-1}(x)=\frac{x+8}{9}[/tex]
C. [tex]f^{-1}(x)=\frac{x-9}{8}[/tex]
D. [tex]f^{-1}(x)=\frac{x-8}{9}[/tex]
Answer (1)
Replace f ( x ) with y : y = 9 x − 8 .
Swap x and y : x = 9 y − 8 .
Solve for y : y = 9 x + 8 .
Replace y with f − 1 ( x ) : f − 1 ( x ) = 9 x + 8 . The answer is f − 1 ( x ) = 9 x + 8 .
Explanation
Understanding the Problem We are given the function f ( x ) = 9 x − 8 and we want to find its inverse, f − 1 ( x ) . The inverse function is found by swapping x and y and then solving for y .
Replace f(x) with y First, replace f ( x ) with y : y = 9 x − 8
Swap x and y Next, swap x and y : x = 9 y − 8
Isolate the term with y Now, solve for y in terms of x . Add 8 to both sides: x + 8 = 9 y
Solve for y Divide both sides by 9: y = 9 x + 8
Write the inverse function Finally, replace y with f − 1 ( x ) : f − 1 ( x ) = 9 x + 8 This matches option B.
Examples
In manufacturing, if f ( x ) represents the cost of producing x items, then f − 1 ( x ) would represent the number of items that can be produced for a cost of x . For example, if the cost of producing x widgets is given by f ( x ) = 9 x − 8 , then the inverse function f − 1 ( x ) = 9 x + 8 tells us how many widgets we can produce for a given cost x . Understanding inverse functions helps in planning and resource allocation.
