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**preview**shows half of the first page. to view the full**2 pages of the document.**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

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(a,b)

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(a,c)

(a,b)

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y

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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.

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