I have a doubt that I imagine to be trivial. How could I establish the image and the inverse image of the function below?
$$\mathit{f}:\mathbb{R}\to\mathbb{R},\mathit{f}(x)=\sin(e^{|x|})$$
I would have to look at your graphical representation? or is there a more immediate way to do this? My question was clear?
You should be able to do it without graphing it. Take it step-by-step. What is the image of $e^{|x|}?$ It's the set $\{e^x\mid x\ge 0\} =[1,\infty).$ What is the range of the $\sin$ function on this interval? It's $[-1,1],$ so that's the image of the composite function.
Now what is the inverse image of the codomain? Write down what this means, and you'll find the answer.