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Lecture

# Lecture 1 - 1.1.pdf

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McGill University

Mathematics & Statistics (Sci)

MATH 140

Ewa Duma

Summer

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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,▯▯o.ing 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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