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

1 Answer
Apr 30, 2016

Answer:

The domain and range are both the whole of #RR#, i.e. #(-oo, oo)#

Explanation:

Any polynomial #f(x)# in #x# is well defined for all #x in RR#, so the domain is #RR#.

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

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

  • As #x# gets large and negative #f(x)# gets large and negative.

  • As #x# gets large and positive #f(x)# gets large and positive.

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

As a result, the graph of #f(x)# will intersect any horizontal line - that is, given any #y in RR#, there is an #x in RR# for which #f(x) = y#.

Hence the range is also the whole of #RR#.

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