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

Answer 1

Abigail Allen

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

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]}

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Savannah Anderson

Domain: All real numbers Range: All real numbers

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.