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

1 Answer
Aug 22, 2015


Domain = #RR# (all real numbers)
Range = #RR# (all real numbers)


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 #oo# in both directions, so the range for this equation is also #RR#, or all real numbers