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

Aug 15, 2017

The domain and range are both the whole of the real numbers $\mathbb{R} = \left(- \infty , \infty\right)$

#### Explanation:

Given:

$f \left(t\right) = {t}^{3} - {t}^{2} + t - 1$

The domain of $f \left(t\right)$ is the set of values of $t$ for which it is well defined.

The given $f \left(t\right)$ is well defined for any value of $t$, so its implicit domain is the whole of the real numbers $\mathbb{R} = \left(- \infty , \infty\right)$.

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

Since $f \left(t\right)$ is a polynomial of odd degree, its range is the whole of the real numbers $\mathbb{R} = \left(- \infty , \infty\right)$.

Note that as $x \to - \infty$ we find $f \left(t\right) \to - \infty$ and as $x \to \infty$ we find $f \left(t\right) \to \infty$. Since $f \left(t\right)$ is continuous, it takes every value in between $- \infty$ and $+ \infty$.

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