Example of onto function R->R^2

[Aziza]

Main Question or Discussion Point

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$\in$ℝ, 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.