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
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-
y x=a y x=a
0 a x 0 a x
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,