Find the inverse of the function [tex]$f(x)=2 x+5$[/tex]. Find [tex]$f^{-1}(x)$[/tex]

To find the inverse of the function f ( x ) = 2 x + 5 , we first replace f ( x ) with y , swap x and y , and solve for y . This gives us the inverse function, f − 1 ( x ) = 2 x − 5 ​ . This inverse can be verified through function composition.
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Answered by Anonymous | 2025-07-04

Replace f ( x ) with y to get y = 2 x + 5 .
Swap x and y to get x = 2 y + 5 .
Solve for y to find y = 2 x − 5 ​ .
Replace y with f − 1 ( x ) , so f − 1 ( x ) = 2 x − 5 ​ ​ .

Explanation

Understanding the Problem We are given the function f ( x ) = 2 x + 5 and we want to find its inverse, denoted as f − 1 ( x ) . The inverse function essentially 'undoes' what the original function does.

Finding the Inverse Function To find the inverse function, we follow these steps:

Replace f ( x ) with y : y = 2 x + 5 .

Swap x and y : x = 2 y + 5 .

Solve for y in terms of x :
x = 2 y + 5
Subtract 5 from both sides:
x − 5 = 2 y
Divide both sides by 2:
y = 2 x − 5 ​

Replace y with f − 1 ( x ) : f − 1 ( x ) = 2 x − 5 ​ .

Verifying the Inverse Function To verify that f − 1 ( x ) = 2 x − 5 ​ is indeed the inverse of f ( x ) = 2 x + 5 , we can check if f ( f − 1 ( x )) = x and f − 1 ( f ( x )) = x .


Let's calculate f ( f − 1 ( x )) :
f ( f − 1 ( x )) = f ( 2 x − 5 ​ ) = 2 ( 2 x − 5 ​ ) + 5 = ( x − 5 ) + 5 = x
Now, let's calculate f − 1 ( f ( x )) :
f − 1 ( f ( x )) = f − 1 ( 2 x + 5 ) = 2 ( 2 x + 5 ) − 5 ​ = 2 2 x ​ = x
Since both compositions result in x , the inverse function is correct.

Final Answer Therefore, the inverse function of f ( x ) = 2 x + 5 is f − 1 ( x ) = 2 x − 5 ​ .

Examples
Imagine you're converting temperatures from Celsius to Fahrenheit using the formula F = 5 9 ​ C + 32 . The inverse function would convert Fahrenheit back to Celsius. This is useful in many real-world scenarios, such as adjusting thermostats or understanding weather reports from different countries. Understanding inverse functions helps you reverse a process and go from the output back to the input, which is a fundamental concept in science and engineering.

Answered by GinnyAnswer | 2025-07-04