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 (2)
Let f − 1 ( 8 ) = y , so f ( y ) = 8 .
Substitute y into f ( x ) = 2 x + 5 to get 2 y + 5 = 8 .
Solve for y : 2 y = 3 , so y = 2 3 .
Therefore, f − 1 ( 8 ) = 2 3 .
Explanation
Understanding Inverse Functions We are given that f ( x ) and f − 1 ( x ) are inverse functions, and f ( x ) = 2 x + 5 . We want to find f − 1 ( 8 ) . The key idea here is to use the property of inverse functions: if f ( x ) = y , then f − 1 ( y ) = x .
Using the Inverse Property Let y = f − 1 ( 8 ) . Then, by the definition of inverse functions, we have f ( y ) = 8 .
Substituting into f(x) We are given the expression for f ( x ) , so we can substitute y into it: f ( y ) = 2 y + 5 .
Setting up the Equation Now we set f ( y ) equal to 8 and solve for y : 2 y + 5 = 8
Isolating the Term with y Subtract 5 from both sides of the equation: 2 y = 8 − 5
2 y = 3
Solving for y Divide both sides by 2 to solve for y :
y = 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 how to find inverse functions helps you reverse the process, going from one scale to the other. This is useful in many real-world situations, such as cooking, weather forecasting, and scientific research, where you might need to switch between different units or scales.
To find f − 1 ( 8 ) given f ( x ) = 2 x + 5 , we set up the equation 2 y + 5 = 8 and solve for y . This results in f − 1 ( 8 ) = 2 3 . Therefore, the answer is 2 3 .
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