What is the inverse of the function [tex]f(x)=-\frac{1}{2}(x+3)[/tex]?
[tex]f^{-1}(x)=[/tex]
$\square$
Answer (1)
Replace f ( x ) with y : y = − 2 1 ( x + 3 ) .
Swap x and y : x = − 2 1 ( y + 3 ) .
Solve for y : y = − 2 x − 3 .
Replace y with f − 1 ( x ) : f − 1 ( x ) = − 2 x − 3 .
Explanation
Understanding the Inverse Function To find the inverse of the function f ( x ) = − 2 1 ( x + 3 ) , we need to switch the roles of x and y and then solve for y . This will give us the inverse function f − 1 ( x ) .
Replacing f(x) with y First, replace f ( x ) with y : y = − 2 1 ( x + 3 ) .
Swapping x and y Next, swap x and y : x = − 2 1 ( y + 3 ) .
Solving for y (Step 1) Now, solve for y . Multiply both sides by − 2 : − 2 x = y + 3 .
Solving for y (Step 2) Subtract 3 from both sides: y = − 2 x − 3 .
The Inverse Function Finally, replace y with f − 1 ( x ) : f − 1 ( x ) = − 2 x − 3 .
Examples
Understanding inverse functions is crucial in many areas, such as cryptography and computer graphics. For example, if a function encodes a message, its inverse decodes it. In computer graphics, inverse functions can help reverse transformations, allowing you to manipulate objects and revert them to their original state. This concept is also used in solving equations, where applying the inverse operation helps isolate the variable you're solving for.
