Exercise 1.4.34

Question

Graph several members of the family of functions

$$
f(x)=\frac{1}{1+a e^{b x}}
$$

where $a>0$. How does the graph change when $b$ changes? How does it change when $a$ changes?

Khagan Gulusoy · Accepted Answer

The larger $b$, the larger the value of $a e^{b x}$. If $a$ is positive, the denominator of the given fraction is positive, and therefore, the function does not have any singularities. For $a=0$, we have:

$$
f(x)=\frac{1}{1+0 \cdot e^{b x}}=\frac{1}{1+0}=1 	ext {. }
$$

Thus, for $a=0$, the function $f$ is constant. For some value $a<0$, the denominator $1+a e^{b x}$ can become 0 . In other words, $f$ has exactly one singularity.

Exercise 1.4.34

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Exercise 1.4.34

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