Question

Y=1/(1+e^x)Can you please give a graph use color for function, asymptotes, etc.A short table of easy pointsThe domain and rangePlease identify and label the asymptotes (Please work this as if you didn’t have a calculator)

Answer 1

`y=(1)/(1+e^x)`

To get a table with values, we have to choose a value for x and calculate the corresponding value for y. For example, choosing x = -5, -2.5, 0, 2.5 and 5.

• x = -5

`y=(1)/(1+e^(-5))\approx0.9933`

• x = -2.5

`y=(1)/(1+e^(-2.5))\approx0.9241`

• x = 0

`y=(1)/(1+e^0)=(1)/(1+1)=(1)/(2)=0.5000`

• x = 2.5

`y=(1)/(1+e^(2.5))\approx0.0759`

• x = 5

`y=(1)/(1+e^5)\approx0.0067`

As we can see, higher values of x get near 0, and lower values of x get near 1. As the logistic functions have the form:

Then we can suppose that those values are the asymptotes. To confirm we have to get the limits when x approximates -∞ and +∞:

`\lim _(x\to-\infty)(1)/(1+e^x)=1`
`\lim _(x\to+\infty)(1)/(1+e^x)=0`

Then our asymptotes are y = 0, 1 and we have no horizontal asymptotes.

Finally, as the x values include from -∞ to +∞, then the domain are all the real values while the range are the values between 0 and 1.

Answer:

• Table

• Asymptotes: ,y = 0, 1 ,and ,x = N/A

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• Domain: ,all real numbers.

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• Range: ,(0, 1)

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• Graph