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Give a simple description of the following functions from $\mathbb{N}$ to $\mathbb{N}$ by stating a rule that specifies $f(x)$ in terms of $x$ for each $x\in \mathbb{N}$. The obvious patterns are intended.

$A= \{(0,0), (1,2), (2,4), (3,6),\ldots\} $

I don't know how to start to be honest.

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    Look at the successive values of the second argument: you get 0,2,4,6, ... surely you see a pattern there! – 2017-02-25
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    Can you show me an example of what you mean? – 2017-02-25
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    start by looking at: $0 \rightarrow 0; 1\rightarrow 2; 2 \rightarrow 4; 3 \rightarrow 6$. Do you see any "rule" jumping out at you. I'm afraid this is an "either you see it or you don't" case. But you really ought to be able to see it. Now write that rule as $f(x) = do-something-to-the-x$. – 2017-02-25
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    OK, the (0,0) means that if x = 0, then f(x) = 0. That is: f(0) = 0. And the (1,2) means that if x = 1, then f(x) = 2. So, f(1) = 2. We also have f(2) = 4, and f(3) = 6 ... do you see what f(x) does to x? – 2017-02-25
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    "Can you show me an example of what you mean?" 0,2,4,6, .... is a pattern. Do you know what the next term is going to be? What about the term after that? What about the $n-th$ term? – 2017-02-25
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    Okay, example. If the pattern were 0->1; 1->2; 2->3; 3->4... I'd see that all we are doing is adding $1$ to each term. So I know $n\rightarrow n+1$ and I'd know the function was $f(x) = x+1$. – 2017-02-25
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    Your right but that's for only one question, because the second question is saying ${(0,0),(1,1),(2,2),(3,3)}$ what about this one, isn't it the same explanation. All I see is adding one too. – 2017-02-25
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    For the *second question* you mention in the comments, $f(0)=0, f(1)=1, f(2)=2,\dots$ I like to think of functions as a machine. The machine takes an input (*often called $x$*) and returns an output which depends on the input (*referred to as $f(x)$*). Here in the second example $1$ gets mapped to $f(1)=1$, so $1$ got sent to $1$. Similarly $2$ gets sent to $2$ and $3$ gets sent to $3$. Although the data given to us was done so using increasing values of $x$, how the machine changed the $x$'s wasn't by adding one. It is okay for the "machine" to be very simple and not change anything. – 2017-02-25

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Your $A= \{(0,0), (1,2), (2,4), (3,6),\ldots\} $ as a function presented as set of ordered pairs is equivalent to

  • $f(0)=0$
  • $f(1)=2$
  • $f(2)=4$
  • $f(3)=6$
  • $\cdots$

and is consistent with the obvious pattern $(n,2n)$, i.e. $f(n)=2n$