Can a function become surjective by restricting its codomain?

[Suyogya]

wikipedia says:
"The exponential function, g: R → R, g(x) = e^x, is not bijective: for instance, there is no x in R such that g(x) = −1, showing that g is not onto (surjective). However, if the codomain is restricted to the positive real numbers R⁺, then g becomes bijective; its inverse is the natural logarithm function ln."
Is altering the set of codomain allowed for a function? if yes then every function would be made surjective just be changing the codomain?
Also by doing this the standard function's definition would get changed

 

[fresh_42]

wikipedia says:
"The exponential function, g: R → R, g(x) = e^x, is not bijective: for instance, there is no x in R such that g(x) = −1, showing that g is not onto (surjective). However, if the codomain is restricted to the positive real numbers R⁺, then g becomes bijective; its inverse is the natural logarithm function ln."
Is altering the set of codomain allowed for a function?

There is no function police! Whether it is allowed or not depends on what you want to do.

If yes then every function would be made surjective just be changing the codomain?

Yes. Every function $f\, : \, M \longrightarrow N$ gets surjective by the restriction $f\, : \, M \longrightarrow \operatorname{im}(f)=f(M) \subseteq N$.

Also by doing this the standard function's definition would get changed

Whether you call this a new function, a restriction of $f$ or don't distinguish them at all is a matter of taste and at best depends on the intentions.