If [tex]g(x)=\frac{5 x-3}{2 x+1}, x \neq \frac{-1}{2}[/tex], find [tex]g^{-1}(x)[/tex] and [tex]g^{-1}(2)[/tex].
Answer (1)
Replace g ( x ) with y and swap x and y : x = 2 y + 1 5 y − 3 .
Solve for y in terms of x : y = 2 x − 5 − x − 3 .
The inverse function is g − 1 ( x ) = 2 x − 5 − x − 3 .
Evaluate g − 1 ( 2 ) : g − 1 ( 2 ) = 2 ( 2 ) − 5 − 2 − 3 = 5 . The final answer is 5 .
Explanation
Understanding the problem We are given the function g ( x ) = 2 x + 1 5 x − 3 and we want to find its inverse g − 1 ( x ) and then evaluate g − 1 ( 2 ) .
Finding the inverse function To find the inverse function, we first replace g ( x ) with y , so we have y = 2 x + 1 5 x − 3 . Next, we swap x and y to get x = 2 y + 1 5 y − 3 . Now we solve for y in terms of x .
Isolating y Multiply both sides of x = 2 y + 1 5 y − 3 by ( 2 y + 1 ) to get x ( 2 y + 1 ) = 5 y − 3 . Expanding the left side gives 2 x y + x = 5 y − 3 .
The inverse function Rearrange the equation to isolate terms with y on one side: 2 x y − 5 y = − x − 3 . Factor out y : y ( 2 x − 5 ) = − x − 3 . Divide by ( 2 x − 5 ) to solve for y : y = 2 x − 5 − x − 3 . Therefore, the inverse function is g − 1 ( x ) = 2 x − 5 − x − 3 .
Evaluating the inverse function Now we need to find g − 1 ( 2 ) . Substitute x = 2 into the expression for g − 1 ( x ) : g − 1 ( 2 ) = 2 ( 2 ) − 5 − 2 − 3 = 4 − 5 − 5 = − 1 − 5 = 5.
Examples
Imagine you are encoding a message using the function g ( x ) . To decode the message, you would need to use the inverse function g − 1 ( x ) . In this case, if the encoded message is 2, then the original message is g − 1 ( 2 ) = 5 . This concept is used in cryptography and data transmission to ensure secure communication.
