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Question 637332: Please help me! What is the domain and range of this polynomial function? x^3+x^2-x-1
Found 2 solutions by MathLover1, jsmallt9:
Answer by MathLover1(20850)   (Show Source):
Answer by jsmallt9(3758)  (Show Source):
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.
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