The inverse of the function [tex]f(x)=\frac{1}{2} x+10[/tex] is shown.
[tex]h(x)=2 x-[/tex]
What is the missing value?
1
5
10
20
Answer (1)
To find the inverse of f ( x ) = 2 1 x + 10 , we set y = 2 1 x + 10 .
Solve for x in terms of y : x = 2 ( y − 10 ) = 2 y − 20 .
The inverse function is h ( x ) = 2 x − 20 .
The missing value is 20 .
Explanation
Understanding the Problem We are given the function f ( x ) = 2 1 x + 10 and its inverse function in the form h ( x ) = 2 x − □ . Our goal is to find the missing value in the inverse function.
Finding the Inverse Function To find the inverse of the function f ( x ) , we first write y = f ( x ) = 2 1 x + 10 . Then, we solve for x in terms of y .
Solving for x We have y = 2 1 x + 10 . Subtracting 10 from both sides gives y − 10 = 2 1 x . Multiplying both sides by 2 gives x = 2 ( y − 10 ) = 2 y − 20 .
Determining the Missing Value Therefore, the inverse function is h ( x ) = 2 x − 20 . Comparing this with the given form h ( x ) = 2 x − □ , we see that the missing value is 20.
Final Answer Thus, the missing value is 20.
Examples
Understanding inverse functions is crucial in many real-world applications. For example, if you have a function that converts Celsius to Fahrenheit, the inverse function converts Fahrenheit back to Celsius. Similarly, in cryptography, encryption and decryption are inverse functions of each other. Knowing how to find and use inverse functions allows us to reverse processes and solve problems in various fields.
