Prove that f(x)=x+e^x is a bijective function.

To prove that a function is bijective, we have to
demonstrate that the function is one-one and on-to
function.

In order to verify if the function is one-one,
we'll apply the first derivative test. This test  verifies if the function is strictly
increasing. If the function is strictly increasing, then the function is an one-one
function.

So, let's calculate  the first
derivative:

f'(x)=1+e^x

1>0
and e^x>0, so, 1+e^x>0 => so the function f(x) is strictly
increasing.

Now, we'll verify if the function is on-to
function. Because f is a sum of elementary function, f is a continuous function
=> f is an on-to function.

We've verified that f is
both, one-one and on-to function, so f(x) is a bijective
function.