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Lecture 1 - 1.1.pdf

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McGill University
Mathematics & Statistics (Sci)
MATH 140
Ewa Duma

MATH140 - Lecture 1 Notes Functions and Models • A function f is a rule that assigns to each element x in a set D exactly one element, called f (x), in a set E. • We consider functions for which the sets D and E are sets of real numbers where D is the domain (x values) of the function reading “f of x” 16 |||| CHAPTER 1 FUNCTIONS AND MODELS • The range is the dependent variable where it is also known as the output (y values) • Four possible ways to represent a function: EXAMPLE 6 Find the domain of each function. • Verbally (description in words) • Numerically (by a table of values) 1 (a) f▯x▯ ▯ s • Visually (graph) (b)t▯x▯ ▯ x ▯ x SOLUTION • Algebraically (explicit formula) N If a function is given by a formul(a) Because the square root of a negative number is not defined (as a real number), domain is not stated explicitly, thethe domain ofsf consists of all values of x such x ▯ 2 ▯ 0 . This is equivalent to that the domain is the set of all nux ▯ ▯2fo, so the domain is the inter▯▯2,▯▯ Example 5 here.* which the formula makes sense and defines a real number. (b) Since 3 A rectangular storage cot▯x▯ ▯r with an▯open top has a volume of 10 m . The length of its base is twice its width. Material for the base costs $10 per square meter; material for the sides costs $6 per square meter. Express the cost of materials as a function of the and dwidth of the base. allowed, we see tht▯x▯ is not defined when x ▯ 0 orx ▯ 1 . Thus the domain of ts Solution: We draw a diagram and▯introduce notation by letting w and 2w be the width and length of the base, respectively, and h be the height. which could also be written in interval notation as The area of the base is (▯▯▯, 0▯ ▯ ▯0, 1▯ ▯ ▯1, ▯▯in dollars, of the materialMfor the base 2 is 10(2w ). Two of the sides have area wh and the other two have area 2wh, so the Thcost of the material for the sides is 6[2(wh) + 2(2wh)]. The total cost is therefore: in thexy-plane are graphs of functions? This is answered by the following test. C = 10(2w ) + 6[2(wh) + 2(2wh)] = 20w + 36wh 2 THE VERTICAL LINE TEST A curve in thexy-plane is the graph of a functionxoif anExpress C as a function of w to get correct answer.e than once. The reason for the truth of the Vertical Line Test can be seen in Figure 13. If each ver- ticalThe Vertical Line Test: A curve in the xy-plane is the graph of a function of x if and only is defined by f▯a▯ ▯ b . But if a lix ▯ a intersects the curve twice, ▯a,b▯and ▯a,c▯, then the curve can’t represent a function because a function can’t assign two different val- ues toa. y x=a y x=a (a,▯c) (a,▯b) (a,▯b) 0 a x 0 a x FIGURE 13 For example, the parabolax ▯ y ▯ 2 shown in Figure 14(a) on the next page is not the graph of a function ofxbecause, as you can see, there are vertical lines that intersect the parabola twice. The parabola, however, does contain the graphs of two functions ofx. 2 2 . Notice that the equation x ▯ y ▯ 2 implies y ▯ x ▯ 2 , so y ▯ ▯ s ▯ 2 Thus the 20 |||| CHAPTER 1 FUNCTIONS AND MODELS 20 |||| CHAPTER 1 FUNCTIONS AND MODELS INCREASING AND DECREASING FUNCTIONS INCREASING AND DECREASING FUNCTIONS The graphshown in FigurSECTION 1.1 FOUR WAYS TO REPRESENT A FUNCTION rises ag||||fr19C The graph shown in Figure 22 rises from A to B, falls from B to C, and rises again from C to D. Thefunction f is saidto be increasingon theinterval ▯a, b▯, decreasingon ▯b, c▯, and y increasingagain on ▯c, d▯. Notice that if 1 and x 2re any two numbers between a and bb, c▯, and SYMMincreasing again on ▯c,Symmetry:ce that if1x and 2 are any two numbers between a and b with x 1▯ x2, then f▯x1▯ ▯ f▯x2▯. We use this as the defining property of an increasing y SYMMETfunction. f satisfies f▯▯x▯ ▯ f▯x▯ for every number x in its domain,then f is called an f(_x) ƒ even function. For instance, the function f▯x▯ ▯ x is even becauseor every number x in its domain, then f is called an If a function f satisfies f▯▯x▯ ▯ f▯x▯ for every number x in its domain, then f is called an _x 0 x x 2 2 f(_x) ƒ even functioy. For instance, the function f▯x▯ 2 B i2 even becauseven because: f▯▯x▯ ▯ ▯▯x▯ ▯ xB▯ f▯x▯ D _x 0 x x D y=ƒ f▯▯x▯ ▯ ▯▯x▯ ▯ x ▯ f▯x▯ The geometric significanceof an even function is that its graph is symmetricwith respect to the y-axis (see Figure 19). This means that if we have plotted the graph of f for x ▯ 0,
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