Example of onto function R->R^2

[Aziza]

What is an example of an "onto" function f: ℝ→ℝ² ?

Any combination of vectors along the x-axis will not be able to leave the x-axis to cover the entire xy plane, so i was thinking of something like f(x) = \left|\stackrel{t}{x}\right| for any t\inℝ, but I was wondering if there are more elegant examples...

 

[Simon Bridge]

See what I can do:

Let: $t\in\mathbb{R}$
... then I can define two functions $x,y \in \mathbb{R}\rightarrow\mathbb{R}\; : \; x=f(t), y=g(t)$
... then I can make $z:\mathbb{R}\rightarrow\mathbb{R}^2\; : \; \vec{z}=(x,y)$

... we would say that f(t) and g(t) is a parameterization of z(x,y).

 

[AKG]

@Aziza, what do you mean by "any t \in \mathbb{R}". If your f is a function, then for a given input x, you need to specify a unique output f(x). E.g. f(1) can only be one point in the plane, you can't have f(1) = (1, 0), and also f(1) = (1,1), and also f(1) = (1, -\pi), and so on.

Since I can't tell whether this is a homework question or not, I'll give you something between a hint and a full answer: for a given x \in \mathbb{R}, split the decimal expansion of x into two parts and create new real numbers out of each of those two parts (we can call these two numbers a(x) and b(x)). Then if you've done things right, the function f:\mathbb{R} \to \mathbb{R}^2 defined by f(x) = (a(x), b(x)) will be surjective.

@Simon, Aziza has asked for a surjective function \mathbb{R} \to \mathbb{R}^2. You've given a function \mathbb{R} \to \mathbb{R}^2, but nothing about it makes it onto. Note that onto is a technical term, another word for it is surjective. A function f: A \to B is surjective iff \forall b \in B, \exists a \in A : f(a) = b.

 

[Simon Bridge]

@AKG: understood, thanks.