What is the domain and range of #f(x) = x^3 - 3x + 2#?

1 Answer
Sep 2, 2015

Domain and range are both #\mathbb{R}#.


The domain is defined as the set of the points which you can give as input to the function. Now, "illegal" operations are:

  1. Dividing by zero
  2. Giving negative numbers to an even root
  3. Giving negative numbers, or zero, to a logarithm.

In your function, there are no denominators, roots or logarithms, so all values can be computed.

As for the range, you can observe that every polynomial #f(x)# with odd degree (in your case the degree is 3), has the following properties:

  1. #\lim_{x \to -\infty} f(x)=-\infty#
  2. #\lim_{x \to +\infty} f(x)= +\infty#

And since polynomials are continuous functions, the range consists in all numbers from #-\infty# to #\infty#, which is to say all the real set.