# How do you find the domain and range of x^5-2x^3+1?

Apr 30, 2016

The domain and range are both the whole of $\mathbb{R}$, i.e. $\left(- \infty , \infty\right)$

#### Explanation:

Any polynomial $f \left(x\right)$ in $x$ is well defined for all $x \in \mathbb{R}$, so the domain is $\mathbb{R}$.

For any polynomial $f \left(x\right)$, as $x$ gets larger, the term with highest degree tends to dominate.

So if $f \left(x\right)$ is of odd degree with positive leading coefficient (as in our example), then:

• As $x$ gets large and negative $f \left(x\right)$ gets large and negative.

• As $x$ gets large and positive $f \left(x\right)$ gets large and positive.

Polynomials are also continuous (no breaks in the graph).

As a result, the graph of $f \left(x\right)$ will intersect any horizontal line - that is, given any $y \in \mathbb{R}$, there is an $x \in \mathbb{R}$ for which $f \left(x\right) = y$.

Hence the range is also the whole of $\mathbb{R}$.

graph{(x^5-2x^3+1-y)(y - 3.2) = 0 [-10, 10, -5, 5]}