If [tex]f(x)=5 x+4[/tex], which of the following is the inverse of [tex]f(x)[/tex]?
A. [tex]f^{-1}(x)=4-5 x[/tex]
B. [tex]f^{-1}(x)=\frac{x-4}{5}[/tex]
C. [tex]f^{-1}(x)=5 x-4[/tex]
D. [tex]f^{-1}(x)=\frac{x-5}{4}[/tex]
Answer (1)
Replace f ( x ) with y : y = 5 x + 4 .
Swap x and y : x = 5 y + 4 .
Solve for y : y = 5 x − 4 .
The inverse function is f − 1 ( x ) = 5 x − 4 .
Explanation
Understanding the Problem We are given the function f ( x ) = 5 x + 4 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 .
Swapping x and y Let y = f ( x ) , so we have y = 5 x + 4 . To find the inverse, we swap x and y to get x = 5 y + 4 .
Isolating the term with y Now we solve for y in terms of x . First, subtract 4 from both sides of the equation: x − 4 = 5 y
Solving for y Next, divide both sides by 5: 5 x − 4 = y
Finding the Inverse Function Thus, the inverse function is f − 1 ( x ) = 5 x − 4 . Comparing this to the given options, we see that it matches option B.
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 back to Celsius. Understanding inverse functions helps in reversing processes or converting back to original values after a transformation.
