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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?

CptSupermrkt
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  • This is the end of the proof. You need to proof, that for every $y\in\mathbb R$, there is some $x\in\mathbb R$, such that $f(x)=y$. That's what you did. – Tomas Jul 07 '13 at 23:38
  • But it's very easy to figure out what x should be in something so simple. In this case just by looking at "-3x + 4" you can "eyeball" it to figure that you need -1/3y + 4/3. But how do you formally arrive at that? – CptSupermrkt Jul 07 '13 at 23:39
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    In a formal proof, you don't need to justify how you found $x$, you just need to show (formally) that $f(x)=y$, maybe show step by step, that this really cancels out. (In order to find that $x$, equating the function with $y$ and solving for $x$ is a good idea in general.) – Tomas Jul 07 '13 at 23:40
  • Ah...so, that really is all there is to it then...No wonder I couldn't find what I was looking for, because I was overthinking it in terms of, "I need to prove every single little step." – CptSupermrkt Jul 07 '13 at 23:43

2 Answers2

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The key word is inverse of a function.

If $y$ represents the value after having $f$ applied on $x$, then we have $$y=-3x+4\,.$$ Now we can imagine that quantities $x$ and $y$ vary together, bound together by the above equation.

To find the inverse function of $f$, we need to express $x$ by means of $y$: $$y-4=-3x\\ 3x=4-y\\ x=\frac{4-y}3\,.$$ Since we could do it in a unique way, $f$ is not only surjective, it is also bijective.

Berci
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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.

Mark Bennet
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