How do you find the inverse of #y = (e^x)/(1+2e^x)#?
To find the inverse of the function (y = \frac{e^x}{1 + 2e^x}), follow these steps:

Replace (y) with (x) and (x) with (y): [x = \frac{e^y}{1 + 2e^y}]

Solve this equation for (y): [x(1 + 2e^y) = e^y]

Expand and rearrange terms: [x + 2xe^y = e^y]

Move terms involving (e^y) to one side: [2xe^y  e^y = x]

Factor out (e^y): [e^y(2x  1) = x]

Divide both sides by (2x  1): [e^y = \frac{x}{2x  1}]

Take the natural logarithm of both sides to isolate (y): [y = \ln\left(\frac{x}{2x  1}\right)]
Therefore, the inverse of the function (y = \frac{e^x}{1 + 2e^x}) is given by (y = \ln\left(\frac{x}{2x  1}\right)).
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Step by step working is shown below.
Use distributive property.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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