If it is not possible to draw a horizontal line across the graph of a function without making contact with the curve representing the function then the function is surjective.
So, $f(x)=x^3$ is a surjectve function. But, $f(x)=x^2$ is not a surjectve function.
Then what about $f(x) = e^x$? Is it a Surjective function?
Why and why not?
It depends. $e^x:\Bbb R\to(0,\infty)$ is a surjection while $e^x:\Bbb R\to\Bbb R$ is not.
According to you definition of "surjectivity", it'd be a nice idea to lookup an image of $f(x)=e^x$. It's kinda like a curve that stays on the upper half plane while never touching the $x$-axis.