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Lecture

# MATH 140 Lecture Notes - Even And Odd Functions, Negative Number, Asteroid Family

Department
Mathematics & Statistics (Sci)
Course Code
MATH 140
Professor
Ewa Duma

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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”
The range is the dependent variable where it is also known as the output (y values)
Four possible ways to represent a function:
Verbally (description in words)
Numerically (by a table of values)
Visually (graph)
Algebraically (explicit formula)
*Prof uses examples from textbook. Showing Example 5 here.*
A rectangular storage container with an open top has a volume of 10 m3. 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
width of the base.
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.
The area of the base is (2w)w = 2w2, so the cost, in dollars, of the material for the base
is 10(2w2 ). Two of the sides have area wh and the other two have area 2wh, so the
cost of the material for the sides is 6[2(wh) + 2(2wh)]. The total cost is therefore:
C = 10(2w2 ) + 6[2(wh) + 2(2wh)] = 20w2 + 36wh
Express C as a function of w to get correct answer.
The 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 one.
EXAMPLE 6 Find the domain of each function.
(a) (b)
SOLUTION
(a) Because the square root of a negative number is not deﬁned (as a real number),
the domain of consists of all values of xsuch that . This is equivalent to
, so the domain is the interval .
(b) Since
and division by is not allowed, we see that is not deﬁned when or .
Thus the domain of is
which could also be written in interval notation as
M
The graph of a function is a curve in the -plane. But the question arises: Which curves
in the -plane are graphs of functions? This is answered by the following test.
THE VERTICAL LINE TEST A curve in the -plane is the graph of a function of if
and only if no vertical line intersects the curve more than once.
The reason for the truth of the Vertical Line Test can be seen in Figure 13. If each ver-
tical line intersects a curve only once, at , then exactly one functional value
is deﬁned by . But if a line intersects the curve twice, at and ,
then the curve can’t represent a function because a function can’t assign two different val-
ues to .
For example, the parabola shown in Figure 14(a) on the next page is not the
graph of a function of because, as you can see, there are vertical lines that intersect the
parabola twice. The parabola, however, does contain the graphs of two functions of .
Notice that the equation implies , so Thus the
upper and lower halves of the parabola are the graphs of the functions
[from Example 6(a)] and .[See Figures 14(b) and (c).] We observe that
if we reverse the roles of and , then the equation does deﬁne as a
function of (with as the independent variable and as the dependent variable) and the
parabola now appears as the graph of the function .h
xyy
xx !h!y"!y2!2yx
t!x"!!sx"2
f!x"!sx"2
y!#sx"2 .
y2!x"2x!y2!2
x
x
x!y2!2
F I G UR E 13
a
x=a
(a,b)
0a
(a,c)
(a,b)
x=a
0x
y
x
y
a
!a,c"!a,b"x!af !a"!b
!a,b"x!a
xxy
xy
xy
!!\$, 0"!!0, 1"!!1, \$"
#x
\$
x"0, x"1%
tx!1x!0t!x"0
t!x"!1
x2!x
!1
x!x!1"
&!2, \$"x% !2
x"2%0f
t!x"!1
x2!x
f!x"!sx"2
16
|| ||
CHAPTER 1 FUNCTIONS AND MODELS
NIf a function is given by a formula and the
domain is not stated explicitly, the convention is
that the domain is the set of all numbers for
which the formula makes sense and deﬁnes a
real number.