Select the correct answer.
Find the inverse of function [tex]f[/tex].
[tex]f(x)=9 x+7[/tex]
A. [tex]f^{-1}(x)=\frac{1}{9} x-\frac{7}{9}[/tex]
B. [tex]f^{-1}(x)=-9 x-7[/tex]
C. [tex]f^{-1}(x)=7 x+9[/tex]
D. [tex]f^{-1}(x)=\frac{7}{9} x-\frac{1}{9}[/tex]
Answer (1)
Replace f ( x ) with y : y = 9 x + 7 .
Swap x and y : x = 9 y + 7 .
Solve for y : y = 9 x − 7 = 9 1 x − 9 7 .
The inverse function is: f − 1 ( x ) = 9 1 x − 9 7 .
Explanation
Understanding the problem We are given the function f ( x ) = 9 x + 7 and we need 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 + 7
Swap x and y Next, swap x and y : x = 9 y + 7
Solve for y Now, solve for y in terms of x . Subtract 7 from both sides: x − 7 = 9 y Divide both sides by 9: y = 9 x − 7 We can rewrite this as: y = 9 1 x − 9 7
Write the inverse function Finally, replace y with f − 1 ( x ) : f − 1 ( x ) = 9 1 x − 9 7 Comparing this with the given options, we see that option A is the correct answer.
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 ) converts Fahrenheit to Celsius. Understanding inverse functions helps in many practical conversion problems.
