Question

What is the domain and range of `t^3-t^2+t-1`?

Answer 1

The domain and range are both the whole of the real numbers `RR = (-oo, oo)`

Explanation:

Given:

`f(t) = t^3-t^2+t-1`

The domain of `f(t)` is the set of values of `t` for which it is well defined.

The given `f(t)` is well defined for any value of `t`, so its implicit domain is the whole of the real numbers `RR = (-oo, oo)`.

The range of `f(t)` is the set of values that it can take for some `t` in the domain.

Since `f(t)` is a polynomial of odd degree, its range is the whole of the real numbers `RR = (-oo, oo)`.

Note that as `x->-oo` we find `f(t)->-oo` and as `x->oo` we find `f(t)->oo`. Since `f(t)` is continuous, it takes every value in between `-oo` and `+oo`.

graph{x^3-x^2+x-1 [-10, 10, -5, 5]}