parametric equations, finding the range of t

Question

When parametrizing a curve how doe we obtain the range of $t$? For example lets say we have the parametrization: $x(t) = 1+3t$ and $y(t) = 2+5t$. How do we find the range of t? $t\to[?,?]$

Answer 1

A note on terminology, as Arbuja has asked about, in mathematics, the term "range" is used to refer to the values coming out of a function, not those that go in. (Further, there are two different meanings in common use for "range", which leads to confustion. So anymore, most people avoid using "range" altogether, calling one meaning "image", and the other "codomain".)

But you are referring the values that are allowed to go into the function. That we call the "domain". So your question should be "what is the domain of this curve?", or "from what domain can $t$ take values". This may seem like pointless nit-picking, but note that the wording of your question has confused Some Math Student and Arbuja to some extent (though I doubt Arbuja was actually much confused, but simply wanted to be sure).

As Some Math Student has said, what domain you pick depends on the curve you are trying to depict. In your example, the curve is a straight line. If you want it to be the entire line, then you would choose the interval $(-\infty, \infty)$ as the domain. If you only wanted the ray with endpoint $(-2, -3)$ and extending into upper right quadrant, you would choose the interval $[-1, \infty)$ as the domain. If you only wanted the line segment with endpoints $(1, 2)$ and $(13, 22)$, then you would choose the interval $[0, 4]$ as the domain.

Which is the correct answer very much depends on what you are needing this curve for.

Answer 2

Eliminating $t$ between the given linear equations we get

$$\frac{x-1}{3}=\frac{y-2}{5}$$

Simplifying,

$$3y=5x +1 $$

This linear functions has $(x,y)$ with domain and range respectively $(-\infty,\infty).$

If for example you had asked with three variables

$$x(t) = 1+3t ,y(t) = 2+5t, z= \cos(t) ;$$

we could refer to $(x,y)$ domains similarly and a range for $z$ as $\pm1$.