(b) Using the method of the equation [tex]$\frac{1+\tanh (x)}{1-\tanh (x)}=e^{2 x}$[/tex], with [tex]$x$[/tex] replaced by [tex]$y$[/tex], let [tex]$y=\tanh ^{-1}(x)$[/tex]. Then [tex]$x=$[/tex]?
Answer (2)
Given y = tanh − 1 ( x ) , express x in terms of y .
Recognize that x = tanh ( y ) .
Therefore, the answer is tanh ( y ) .
Explanation
Problem Setup We are given the equation 1 − t a n h ( x ) 1 + t a n h ( x ) = e 2 x and asked to replace x with y . We are also given that y = tanh − 1 ( x ) . The goal is to find an expression for x in terms of y .
Expressing x in terms of y Since y = tanh − 1 ( x ) , we can express x as the hyperbolic tangent of y , which is x = tanh ( y ) .
Substituting y for x Now, we substitute y for x in the given equation: 1 − t a n h ( y ) 1 + t a n h ( y ) = e 2 y . This equation relates the hyperbolic tangent of y to the exponential function.
Substituting x for tanh(y) Since x = tanh ( y ) , we can substitute x for tanh ( y ) in the equation 1 − t a n h ( y ) 1 + t a n h ( y ) = e 2 y to get 1 − x 1 + x = e 2 y .
Taking the Natural Logarithm Taking the natural logarithm of both sides of 1 − x 1 + x = e 2 y , we get ln ( 1 − x 1 + x ) = 2 y .
Solving for y Solving for y , we have y = 2 1 ln ( 1 − x 1 + x ) . This confirms that y = tanh − 1 ( x ) = 2 1 ln ( 1 − x 1 + x ) .
Final Answer The question asks for x = □ . Since y = tanh − 1 ( x ) , then x = tanh ( y ) . Therefore, the answer is tanh ( y ) .
Examples
In signal processing, the hyperbolic tangent function is used to model the activation function of neurons in neural networks. If you know the inverse hyperbolic tangent of a signal, you can use the hyperbolic tangent function to find the actual signal value. This is crucial in designing and analyzing neural networks for various applications like image recognition and natural language processing.
To express x in terms of y , we use the relationship derived from the inverse hyperbolic tangent function. Since y = tanh − 1 ( x ) , it follows that x = tanh ( y ) . Therefore, the final answer is x = tanh ( y ) .
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