Find the inverse of [tex]f(x)=\frac{-2 x+1}{3 x+2}[/tex]
[tex]f^{-1}(x)=[/tex]
Answer (2)
Swap x and y in the original equation: x = 3 y + 2 − 2 y + 1 .
Multiply both sides by 3 y + 2 : x ( 3 y + 2 ) = − 2 y + 1 .
Solve for y : y = 3 x + 2 1 − 2 x .
The inverse function is f − 1 ( x ) = 3 x + 2 1 − 2 x .
Explanation
Problem Analysis We are given the function f ( x ) = f r a c − 2 x + 1 3 x + 2 and we want to find its inverse f − 1 ( x ) . The inverse function is found by swapping x and y and solving for y .
Swapping x and y Let y = f ( x ) , so we have y = f r a c − 2 x + 1 3 x + 2 . To find the inverse, we swap x and y to get x = f r a c − 2 y + 1 3 y + 2 . Now we solve for y .
Multiply by the denominator Multiply both sides by 3 y + 2 to get x ( 3 y + 2 ) = − 2 y + 1 .
Expanding the equation Expand the left side: 3 x y + 2 x = − 2 y + 1 .
Rearranging terms Move all terms containing y to the left side and all other terms to the right side: 3 x y + 2 y = 1 − 2 x .
Factoring out y Factor out y : y ( 3 x + 2 ) = 1 − 2 x .
Isolating y Divide by 3 x + 2 to isolate y : y = f r a c 1 − 2 x 3 x + 2 .
The Inverse Function Therefore, the inverse function is f − 1 ( x ) = f r a c 1 − 2 x 3 x + 2 .
Examples
In cryptography, inverse functions can be used to decode messages. If a message is encoded using a function, the inverse function can be used to decode it back to its original form. For example, if the encoding function is f ( x ) = f r a c − 2 x + 1 3 x + 2 , then the decoding function would be its inverse, f − 1 ( x ) = f r a c 1 − 2 x 3 x + 2 . This ensures that the original message can be recovered accurately.
The inverse of the function f ( x ) = 3 x + 2 − 2 x + 1 is f − 1 ( x ) = 3 x + 2 1 − 2 x . This is determined by swapping x and y, rearranging the equation, and solving for y. Following these steps leads to the correct inverse function.
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