# Domain of an exponential function (US Common Core, High School Math)

I came across the question below on IXL.com. (I have reproduced it in its entirety. Given were 5 possible answers to choose from.)

It strikes me that there is not one single correct answer for this question. But perhaps according to the US Common Core standard there is. If so, what is the single correct answer and why? And if not, would this question simply be wrong? • The correct answer is all real numbers. But, perhaps, could you list the answers they give? Mar 11 '15 at 19:09
• Personally, I think the graph is inappropriately misleading, because it looks to me like what someone might draw to indicate the $y$-axis is a vertical asymptote. If the graph came with an equation, I'd feel much better about it. Mar 11 '15 at 20:15
• @DaveLRenfro absolutely. I think math tests are riddled with this kind of ambiguity. How can we assume it crosses or doesn't cross the Y axis? At least the Y=4 asymptote is clear, if not unhelpful. Mar 12 '15 at 0:07
• @JoeTaxpayer: you mean if unhelpful, right? =P Jan 7 '17 at 23:59
• 'Clear, even if not helpful." Is what I think I meant to write. Jan 8 '17 at 0:40

The correct answer is obviously all reals, since it is an exponential function. I think the big point trying to be made here is that exponential functions are defined anywhere, so even though the graph is (more than likely very purposefully) misleading to show that there appears to be an asymptote at x=0, The key is relying on the student's ability to reason that exponential functions all look like

$f(x)= a^{(x-c)} +d$ where a,c,d $\in \mathbb R$ and that there just is not a value that you can not plug in to the function.

That being said, the graph is horribly drawn, and I had a similar problem with my students when we started working with exponential functions because of the fact that no matter the window you use the graph just always seems to sort of stop. That's why I used Desmos along with the question of "do you think it stops at [whatever value is at the top of the window]? " I would zoom out or pan up to show them that the graph always sort of went on, but it grew very quickly, which is why it would look like it had an asymptote at some x-value.

It is a poorly written question because it presumes there is some sort of universal definition of what is meant by an "exponential function."

You can closely replicate the graph with $f(x)=3+e^{-1/x}$ with $x<0$.

There is consensus about what we mean by polynomial and rational functions. But depending on the context, it would not be unreasonable to conceptualize an "exponential" function as being any function in the field of functions that we get by adjoining exp to the field of rational functions.

In the context of the original question, one can guess that what is meant by an "exponential function" is something of the form $\hbox{polynomial}+Ae^{cx}$ or perhaps $\hbox{rational}+Ae^{cx}$

• But. The problem is really to acknowledge we are looking at an 'all reals' domain so setting X less than 0 is cheating. And the equation doesn't offer the fast flatline toward 4. Mar 13 '15 at 23:49

Because I'm a bit OCD, I had to know. @Jhecht gave enough hints for me to push for a solution. $y=10^{\left(x+1\right)}+4$

is as close at it gets to what the real equation is. And the equation makes it obvious the Y-intercept is 14.

And while it's fair enough at some level to expect an understanding that this function will have all reals as domain, I still find the graph misleading. I remain curious what the choices were for answers.

(On further reflection, I have (-1,5) as a point, the OP graph is closer to (-1,5.5) so the exponent is closer to x+1.1. The point remains, the Y intercept is high)