What is the domain and range of y=x^3-x?

Aug 22, 2015

Domain = $\mathbb{R}$ (all real numbers)
Range = $\mathbb{R}$ (all real numbers)

Explanation:

The domain of a function is a "set" of all of the x-values for which there is a corresponding y-value. In this case, since we just have a simple binomial with no denominator or any way to get an undefined answer, you should be able to plug any number you want for x and get a real "y" answer. But there may be a limit to what "y" can be. Let's look at this graph.

The first (and foremost) thing to notice is the ${x}^{3}$ term. As you may be able to tell, this term will grow a lot faster than the regular $x$. This means that as you zoom further and further out, the graph of this equation will start to look a lot like the graph of $y = {x}^{3}$. This is a general rule: The highest degree (exponent) term will grow the fastest, so the other terms will not determine the range.

The graph of $y = {x}^{3}$ is a familiar one.graph{x^3 [-10, 10, -5, 5]}
As you can see, the y-values don't have any sort of peak. They go off towards $\infty$ in both directions, so the range for this equation is also $\mathbb{R}$, or all real numbers