# How do you find the inverse of #y = e^x/(1 + 4 e^x)#?

To find the inverse of ( y = \frac{e^x}{1 + 4e^x} ), first, interchange the roles of ( x ) and ( y ), then solve for ( y ) in terms of ( x ).

- Start with the original equation: ( y = \frac{e^x}{1 + 4e^x} ).
- Swap ( x ) and ( y ): ( x = \frac{e^y}{1 + 4e^y} ).
- Multiply both sides by ( (1 + 4e^y) ): ( x(1 + 4e^y) = e^y ).
- Distribute and rearrange terms: ( x + 4xe^y = e^y ).
- Move all terms involving ( e^y ) to one side: ( e^y - 4xe^y = x ).
- Factor out ( e^y ): ( e^y(1 - 4x) = x ).
- Solve for ( e^y ): ( e^y = \frac{x}{1 - 4x} ).
- Take the natural logarithm of both sides: ( y = \ln\left(\frac{x}{1 - 4x}\right) ).

Therefore, the inverse of the function ( y = \frac{e^x}{1 + 4e^x} ) is ( y = \ln\left(\frac{x}{1 - 4x}\right) ).

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This question would be a "solving for the inverse of a rational functions question" and you would follow the same standard procedure as you would for solving those equations.

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