does the inverse of the following function exist? If not (as said by many calculators), I'd like to know why, plotting it in desmos seems to show it is one-to-one.
$$f(x)=9^x-e^x$$
I found a similar question from 7 years ago,
but it did not result in anything clear.
Maybe a more specific question might help solve this doubt I've had for the last 2 or 3 weeks.
If you only want to know if the inverse exists, you should check if the function is bijective. Doing so graphically is quite easy using the "horizontal line check" for injectivity. If you graph $f(x)=9^x-e^x$, you will see that there exists at least one horizontal line that passes through two points of the graph. This implies that $f$ is not injective, and thus, it's not invertible. You could also try and prove this algebraically.
Edit: In your original post, you mention the sum or difference of two or more exponentials. If $f$ is a sum of exponentials with bases $b>1$, I believe it will always be bijective due to it being a sum of increasing functions. However, with a difference of exponentials, I believe the function might (almost) never be one to one. Also, you mentioned that graphing through Desmos seems to show $f$ is one-to-one. However, note that using $y=-0.15$ we get a counterexample using the aforementioned test.
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