Question

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

Answer 1

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]}