Stitz-Zeager_College_Algebra_e-book

9 show that the end behavior of a linear function f x

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Unformatted text preview: be continuous where it has a ‘break’ or ‘hole’ in the graph; everywhere else, the function is continuous. The function is continuous at the ‘corner’ and the ‘cusp’, but we consider these ‘sharp turns’, so these are places where the function fails to be smooth. Apart from these four places, the function is smooth and continuous. Polynomial functions are smooth and continuous everywhere, as exhibited in graph on the right. ‘hole’ ‘corner’ ‘cusp’ ‘break’ Pathologies not found on graphs of polynomials The graph of a polynomial The notion of smoothness is what tells us graphically that, for example, f (x) = |x|, whose graph is the characteric ‘∨’ shape, cannot be a polynomial. The notion of continuity is what allowed us to construct the sign diagram for quadratic inequalities as we did in Section 2.4. This last result is formalized in the following theorem. Theorem 3.1. The Intermediate Value Theorem (Polynomial Zero Version): If f is a polynomial where f (a) and f (b) have different signs, then f has at least one zero between x = a and x = b...
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This note was uploaded on 05/03/2013 for the course MATH Algebra taught by Professor Wong during the Fall '13 term at Chicago Academy High School.

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