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

$y=\frac{1}{1+e^x}$

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

$y=\frac{1}{1+e^{-5}}\approx0.9933$

• x = -2.5

$y=\frac{1}{1+e^{-2.5}}\approx0.9241$

• x = 0

$y=\frac{1}{1+e^0}=\frac{1}{1+1}=\frac{1}{2}=0.5000$

• x = 2.5

$y=\frac{1}{1+e^{2.5}}\approx0.0759$

• x = 5

$y=\frac{1}{1+e^5}\approx0.0067$

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}\frac{1}{1+e^x}=1$$\lim _{x\to+\infty}\frac{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