You can
put this solution on YOUR website!The domain is all values that
can take on. In your case:
the domain: all real numbers
Since in your case you will eventually cover all possible values of
(for all
you will have an
), then the range is also "all real numbers".
so,
the range: all real numbers
You can
put this solution on YOUR website!By definition the exponents in a polynomial are whole numbers (0, 1, 2, 3, ...). Since
every real number can be raised to a whole number power, the domain of
all polynomials, including this one, is the set of all real numbers.
The range of a polynomial is determined by the term with the highest degree. If the degree of the highest power term is odd then the range is all real numbers (because
every real number has an odd-numbered root (no matter which odd number the root is). Since the highest degree term of your polynomial is
and its degree, 3, is odd, then your polynomial has a range of all real numbers.
If the degree of the polynomial is even then the range will not be all real numbers. This is so because not all real numbers have even-numbered roots. Only the non-negative real numbers have even-numbered roots. For example, there is no real number that is the square root of -14. So there is no way for
to turn out to be a -14. A general description of ranges for polynomials with an even degree:
- If the leading coefficient (coefficient of the highest power term) is positive, then there will be a lower limit but no upper limit. IOW, The range is all real numbers greater than or equal to some minimum value.
- If the leading coefficient is negative, then there will be an upper limit but no lower limit. IOW, The range is all real numbers less than or equal to some maximum value.
(