If [tex]$f(x)$[/tex] and [tex]$f^{-1}(x)$[/tex] are inverse functions of each other and [tex]$f(x)=2 x+5$[/tex], what is [tex]$f^{-1}(8)$[/tex]?
Answer (1)
Recognize that finding f − 1 ( 8 ) means finding x such that f ( x ) = 8 .
Set up the equation 2 x + 5 = 8 .
Solve for x by subtracting 5 and dividing by 2.
Conclude that f − 1 ( 8 ) = 2 3 .
Explanation
Understanding the Problem We are given that f ( x ) and f − 1 ( x ) are inverse functions, and f ( x ) = 2 x + 5 . We want to find the value of f − 1 ( 8 ) .
Using Inverse Function Property Since f ( x ) and f − 1 ( x ) are inverse functions, we know that if f ( a ) = b , then f − 1 ( b ) = a . Therefore, to find f − 1 ( 8 ) , we need to find the value of x such that f ( x ) = 8 .
Setting up the Equation We set f ( x ) = 8 and solve for x :
2 x + 5 = 8
Isolating the Term with x Subtract 5 from both sides of the equation: 2 x = 8 − 5
Simplifying 2 x = 3
Solving for x Divide both sides by 2: x = 2 3
Finding the Inverse Value Therefore, f − 1 ( 8 ) = 2 3 .
Examples
Imagine you're converting temperatures between Celsius and Fahrenheit. If f ( x ) converts Celsius to Fahrenheit, then f − 1 ( x ) converts Fahrenheit back to Celsius. Knowing that f ( x ) = 5 9 x + 32 , if you want to find the Celsius temperature that corresponds to 68°F, you would calculate f − 1 ( 68 ) . This concept of inverse functions is crucial in many real-world conversions and calculations, from currency exchange to unit conversions in physics and engineering.
