(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] ______ [tex]$\tanh (y)$[/tex], so we have the following. [tex]$e^{2 y}=\frac{1+\tanh (y)}{\square}=\frac{1+x}{1-x}$[/tex]
Answer (1)
tanh ( y ) 1 − 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 find the missing expressions when x is replaced by y and y = tanh − 1 ( x ) .
Finding the first missing expression Since y = tanh − 1 ( x ) , it follows that x = tanh ( y ) . Thus, the first blank should be filled with tanh ( y ) .
Finding the second missing expression Replacing x with y in the given equation 1 − t a n h ( x ) 1 + t a n h ( x ) = e 2 x , we get 1 − t a n h ( y ) 1 + t a n h ( y ) = e 2 y . We are also given that e 2 y = 1 − x 1 + x . Comparing these two equations, we can see that the missing expression in the denominator is 1 − tanh ( y ) .
Final Equation Since x = tanh ( y ) , we can rewrite the missing expression as 1 − x . Therefore, the equation becomes e 2 y = 1 − t a n h ( y ) 1 + t a n h ( y ) = 1 − x 1 + x .
Examples
In signal processing, the hyperbolic tangent function and its inverse are used in designing filters and analyzing signals. Understanding the relationship between tanh ( x ) and e 2 x can help engineers simplify complex expressions and design efficient systems. For example, if you are designing a communication system, you might use these relationships to model the behavior of a nonlinear amplifier or to optimize the performance of a modulation scheme.
