SOLUTION: what would the graph of an exponential function with range y<1,y∈R and a y-intercept of 0 look like. And what are the domain, horizontal asymptote, x-intercept.

We want a graph like this. 



One that goes through the origin to have a y=intercept of 0. 
It must go up, as close to 1 as possible, but never quite to 1,
because it has a range of y < 1, so it has a horizontal asymptote at y=1

How do we get an equation for a graph to look like that?

We start with the basic exponential equation.
. It has a range of x > 0, and horizontal asymptote y=0,
which is the x-axis


We want to get it below the x-axis, so we reflect it in the x-axis by
multiplying the right side by -1





Now all we need to do is shift the graph and its horizontal asymptote
2 units upward, by adding +1 to the right side. That will take care of
both raising the asymptote from y=0 to y=1 and the y-intercept of -1 up to 0.




Its domain is 

Its horizontal asymptote is 

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

Edwin

In his post, Edwin, actually, found a function  y = , satisfying the imposed conditions

    



But he mistakenly wrote it as  y = .

Here "2" is a mistake or a typo. The correct formula for the function is  y = .


    Its domain is  .

    Its horizontal asymptote is  .

    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 = , with positive real coefficient "a" in the exponent.


    


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  .

    Their horizontal asymptote is  .

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



There are solutions of another form. 
They are of the form  y = , with NEGATIVE real coefficient "a" in the exponent.
They also satisfy all imposed conditions.


See the plots below

    


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  .

    Their horizontal asymptote is  .

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

Ikleyn is right that I typed 2 for 1, but I could not have 
shown the graph with 2 in there using the site's notation 
for graphing.  You'll notice I did correct the typo above.

Also the request was for the graph of AN exponential 
function, not THE ONE AND ONLY possible one.  

There is such a thing as over-teaching by over-
generalizing. Students who post on here are 
struggling with their math class, not doing 
mathematical research.  Sometimes I think some 
tutors on here are not trying to teach students,
but instead, putting on a show to "say without 
saying" to other tutors, "Look how clever I am!
 -- (and how unclever you are!)"
 
Edwin