Find the inverse of [tex]$f(x)=\frac{-3 x+3}{x-5}$[/tex]
Answer (2)
Replace f ( x ) with y : y = x − 5 − 3 x + 3 .
Swap x and y : x = y − 5 − 3 y + 3 .
Solve for y : y = x + 3 5 x + 3 .
Replace y with f − 1 ( x ) : f − 1 ( x ) = x + 3 5 x + 3 . The inverse function is f − 1 ( x ) = x + 3 5 x + 3 .
Explanation
Problem Analysis We are given the function f ( x ) = x − 5 − 3 x + 3 and we want to find its inverse, which we denote as f − 1 ( x ) .
Steps to Find the Inverse To find the inverse of a function, we follow these steps:
Replace f ( x ) with y : y = x − 5 − 3 x + 3
Swap x and y : x = y − 5 − 3 y + 3
Solve for y in terms of x .
Solving for y Let's solve for y :
x = y − 5 − 3 y + 3
Multiply both sides by ( y − 5 ) :
x ( y − 5 ) = − 3 y + 3
Expand the left side:
x y − 5 x = − 3 y + 3
Move all terms containing y to the left side and all other terms to the right side:
x y + 3 y = 5 x + 3
Factor out y from the left side:
y ( x + 3 ) = 5 x + 3
Divide both sides by ( x + 3 ) to isolate y :
y = x + 3 5 x + 3
The Inverse Function Replace y with f − 1 ( x ) :
f − 1 ( x ) = x + 3 5 x + 3
Final Answer Therefore, the inverse of the function f ( x ) = x − 5 − 3 x + 3 is f − 1 ( x ) = x + 3 5 x + 3 .
Examples
In cryptography, inverse functions can be used for encoding and decoding messages. If f ( x ) represents an encoding function, then f − 1 ( x ) would be the decoding function. For example, if f ( x ) = x − 5 − 3 x + 3 is used to encode a message, the recipient would use f − 1 ( x ) = x + 3 5 x + 3 to decode it back to the original message. This ensures secure communication where only the intended recipient can decipher the message.
The inverse of the function f ( x ) = x − 5 − 3 x + 3 is f − 1 ( x ) = x + 3 5 x + 3 . To find it, we swapped the variables and solved for y . This gives us the function that reverses the operation of the original function.
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