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Properties of Functions New Functions from Old
Properties of Functions New Functions from Old

Document
Document

... ( Weightage of Marks : 80% ) Special credit will be given for accuracy and relevance in the answer. Marks will be deducted for incorrect spelling, untidy work and illegible handwriting. The marks for each question has been indicated in the margin. ...
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... mean it crosses the x-axis at that point. • The function may just touch the x-axis at that point: it is still a zero but does not cross. • Look at y = x2 + 6x + 9 • It factors (x + 3)(x + 3) and just touches at x = -3 • You can have more than one factor the same • Factor: y = x4 – 4x3 – 2x2 + 12x + ...
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Algebra 1 Lesson 5

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Topology I – Problem Set Two Fall 2011

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Midterm Practice Test

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1 Review Sheet 1. Basic Concepts A polynomial is an expression in

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THE UNIVERSITY OF TOLEDO Topology M.A. Comprehensive Examination April , 2010

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... Suppose not. Then the open sets Uj = X − Aj cover X. Since X is compact, only finitely many are needed, so X = U1 ∪ · · · ∪ Un for some n. But since Uj ⊂ Uj+1 , this implies that X = Un . However, X − Un = An 6= ∅, so this is a contradiction. 5. Let U ⊂ R be an open set. Prove that U is a countable ...
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Continuous function

In mathematics, a continuous function is, roughly speaking, a function for which small changes in the input result in small changes in the output. Otherwise, a function is said to be a discontinuous function. A continuous function with a continuous inverse function is called a homeomorphism.Continuity of functions is one of the core concepts of topology, which is treated in full generality below. The introductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbers. In addition, this article discusses the definition for the more general case of functions between two metric spaces. In order theory, especially in domain theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article.As an example, consider the function h(t), which describes the height of a growing flower at time t. This function is continuous. By contrast, if M(t) denotes the amount of money in a bank account at time t, then the function jumps whenever money is deposited or withdrawn, so the function M(t) is discontinuous.
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