Section 1.1 through 1.3 (Notes).PDF

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MATH 1500

Sections 1.1 -1.3 Recall that a function is a triplet D;E;f, where D is a set called domain, E is a set called codomain and f is a rule/law which asso- ciates to each element x in D one and only one element in E, denoted by f(x). Notation: f : A ! B \f is a function with domain A and codomain B" x 2 A \x belongs to the set A" Example: x 2 (▯1;+1) denotes \x belongs to the real line" The set of natural numbers 1;2;3;::: is denoted by N, the set of integers (that is, 0;1;2;3;::: and ▯1;▯2;▯3;:::) by Z, the set of rational numbers by Q, and the set of real numbers by R. Convention: if no domain is speci▯ed for a function given by an al- gebraic expression, then the domain of that function is the largest set of real numbers for which the expression is meaningful. Problem 1. In each case determine the domain of the function. p 2 (a) g(x) = 1 ▯ x 5 (b) f(x) = 2x + 7x ▯ 1 x ▯ 2x + 6 p p (c) g(x) = 32 ▯ x 1 The range of a function f is the set ff(x) : x is in the domain of fg. Thegraphofa functionf isthe setf(x;f(x)) : x is in the domain of fg. Vertical line test. A curve in the xy-plane is the graph of a function of x if and only if no vertical line intersects the curve more than once. Examples ... Problem 2. Sketch the graph of the given function. (a) f(x) = jxj ( (b) g(x) = ▯x + 1; x < ▯2 x ▯ 4; x ▯ ▯2 2 De▯nition 1. A function f is called increasing on an interval I if f(x 1 < f(x )2 whenever x < 1 2 in I: It is called decreasing on an interval I if f(x ) > f(x ) whenever x < x in I: 1 2 1 2 Examples ... De▯nition 2. A function f is called even if f(▯x) = f(x) for every x in the domain of f: It is called odd if f(▯x) = ▯f(x) for every x in the domain of f: Examples ... 3 Important classes of functions. n
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