Question 1206248
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        The goal of this my post is twofold.

        First is to make a correction in Edwin's post.

        Second is to expand it and to show another examples of similar functions.



<pre>
In his post, Edwin, actually, found a function  y = {{{1-e^x}}}, satisfying the imposed conditions

    {{{graph(200,200,-3,3,-3,3, -e^(x)+1,sqrt(sin(9x))/sqrt(sin(9x))))}}}



But he mistakenly wrote it as  y = {{{2-e^x}}}.

Here "2" is a mistake or a typo. The correct formula for the function is  y = {{{1-e^x}}}.


    Its domain is  {{{(matrix(1,3,-infinity,",",infinity))}}}.

    Its horizontal asymptote is  {{{y=1}}}.

    Its x-intercept (and its y-intercept) is (0,0). 


Actually, there are infinitely many of such functions, satisfying the imposed conditions.

They are of the form  y = {{{1 - e^(ax)}}}, with positive real coefficient "a" in the exponent.


    {{{graph(200,200,-3,3,-3,3, -e^(x)+1, -e^(2x)+1, -e^(0.5x)+1,
       sqrt(sin(9x))/sqrt(sin(9x))))}}}


In this plot, red curve is for a = 1; green curve is for a = 2 and blue curve is for a = 0.5.


    Their domain is  {{{(matrix(1,3,-infinity,",",infinity))}}}.

    Their horizontal asymptote is  {{{y=1}}}.

    Their x-intercept (and their y-intercept) is (0,0). 



There are solutions of another form. 
They are of the form  y = {{{1 - e^(ax)}}}, with NEGATIVE real coefficient "a" in the exponent.
They also satisfy all imposed conditions.


See the plots below

    {{{graph(200,200,-3,3,-3,3, -e^(-x)+1, -e^(-2x)+1, -e^(-0.5x)+1,
             sqrt(sin(9x))/sqrt(sin(9x))))}}}


In this plot, red curve is for a = -1; green curve is for a = -2 and blue curve is for a = -0.5.


    Their domain is the same  {{{(matrix(1,3,-infinity,",",infinity))}}}.

    Their horizontal asymptote is  {{{y=1}}}.

    Their x-intercept (and their y-intercept) is (0,0). 
</pre>

Solved.