Can anyone provide me a step-by-step proof for proving a function IS onto/surjective? I've seen the definition, I've seen several examples and anti-examples (e.g. the typical x squared example).  I get the idea, but I can't seem to find a proof for proving that a function IS onto, with proper explanation start to finish.
Given:


*

*$f: R$ $\rightarrow$ $R$ 

*$f(x) = -3x + 4$


Prove that the above function is "onto."  I know that this IS onto, but what would a dry, stone cold proof look like for this, that says like Step 1 with justification, step 2 with justification, and so on?
The closest thing I could find to what I'm looking for: http://courses.engr.illinois.edu/cs173/sp2009/lectures/lect_15_supp.pdf in Section 3.  It says to prove that g(x) = x - 8 is onto, and it does so by setting x to be (y + 8).  But...why choose that value?  What formula or strategy is there for determining what x should be?  It appears as though you want x to be whatever will get rid of other stuff (+8 against a -8).
So with some basic algebra, I think I can set x to $-\frac13$y + $\frac43$.  And this is valid by the definition of real numbers, yes?  This properly cancels everything out so that f(x) = y.  Is that really the end of the proof?.....or am I way off the track?

What you need to do to prove that a function is surjective is to take each value $y$ and find - any way you can - a value of $x$ with $f(x)=y$. If you succeed for every possible value of $y$, then you have proved that $f$ is surjective.
So we take $x=-\cfrac 13y+ \cfrac 43$ as you suggest. This is well-defined (no division by zero, for example)
Then $f(x)=-3x+4=-3\left(-\cfrac 13y+ \cfrac 43\right)+4=y-4+4=y$
So your formula covers every $y$ at once. And because you have covered every $y$ you can say that you have a surjection.
