Chapter 1 Functions and Linear Models Sections 1.1 and 1.2

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Chapter 1Functions andLinear ModelsSections
1.1 and 1.2
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Functions and Linear Models

• Functions Numerical, Algebraic
• Functions Graphical
• Linear Functions
• Linear Models

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Functions
A real-valued function f is a rule that assigns
to each real number x in a set X of numbers, a
unique real number y in a second set Y of
numbers. The set X is called the domain of the
function f and the second set Y is called the
codomain of f.
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Functions
For each element x in the domain X of the
function, the corresponding element y in Y is
called the image of x under the function f. The
image is denoted by f (x), that is, y f (x).
f (x) is read f of x. The set of all images
of the elements of the domain is called the range
of the function.
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A way to picture a function is by an arrow diagram
f
x
y
x
y
x
X
Y
DOMAIN
RANGE
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Functions

• A symbol that represents an arbitrary number in
  the
• domain of a function f is called an independent
• variable. A symbol that represents a number in
  the
• range of f is called a dependent variable.
• A function can be specified
• algebraically by means of a formula
• numerically by means of a table
• graphically by means of a graph

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Note on Domains

• The domain of a function is not always specified
• explicitly.
• If no domain is specified for the function f, we
  take
• the domain to be the largest set of numbers x
  for
• which f (x) makes sense.
• This "largest possible domain" is sometimes
• called the natural domain.

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Algebraically Defined Function
Is a function represented by a formula. It has
the format y f (x) expression in x
is a function.
Example
Substitute 5 for x
Substitute xh for x
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Algebraically Defined Function
Is a function represented by a formula. It has
the format y f (x) expression in x
is a function.
Example
In this case the natural domain of the function
is the set of all real numbers. That is, Dom f
(? , ?)
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Algebraically Defined Function
is a function.
Example
In this case the natural domain of the function
is the set
In interval notation this is
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Algebraically Defined Function
is a function.
Example
In this case the natural domain of the function
consists of all values of z such that
In interval notation this is
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Numerically Specified Function
This is the case when we give numerical values
for the function (the outputs, say the y-values)
for certain values of the independent variable,
say x. In this case the function is represented
by a table which looks like.
x-values x1 x2 xn
y f (x) f (x1) f (x2) f (xn)
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Numerically Specified Function
Example Suppose that the function f is specified
by the following table.
x 0 1 2 3.7 4
f (x) 3.01 -1.03 2.22 0.01 1
Then, f (0) is the value of the function when x
0. From the table, we obtain f (0) 3.01
Look on the table where x 0 f (1) -1.03
Look on the table where x 1 and so on
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Numerically Specified Function
Example The human population of the world P
depends on the time t. The table gives estimates
of the world population P (t) at time t, for
certain years. For instance, However, for each
value of the time t, there is a corresponding
value of P, and we say that P is a function of t.
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Numerically Specified Function
Example The data represents the velocity V of an
object, in feet/sec, after t seconds have elapsed.
t 0 1 2 3 4
V(t) 2.2 3.55 4.9 6.25 7.6
Note at 2 seconds the object is going at 4.9
ft/sec, that is V(2) 4.9 ft/sec. The table can
be represented graphically as follows
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Numerically Specified Function
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Mathematical Modeling
To mathematically model a situation means to
represent it in mathematical terms. The
particular representation used is called a
Mathematical model of the situation. Mathematical
models do not always represent a situation
perfectly or completely. Some represent a
situation only approximately, whereas others
represent only some aspects of the situation.
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Mathematical Modeling
Example The monthly payment, M, necessary to
repay a home loan of P dollars, at a rate of r
per year (compounded monthly), for t years, can
be found using
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Mathematical Modeling
Example A farmer has 1000 yards of fencing to
enclose a rectangular garden. Express the area A
of the rectangle as a function of the width x of
the rectangle. What is the domain of A?
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Mathematical Modeling
Example Human population The table shows data
for the population of the world in the 20th
century. The figure shows the corresponding
scatter plot.
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Mathematical Modeling
The pattern of the data points suggests
exponential growth.
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Mathematical Modeling
We use a graphing calculator with exponential
regression capability to apply the method of
least squares and obtain the exponential model
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Mathematical Modeling
We see that the exponential curve fits the data
reasonably well. The period of relatively slow
population growth is explained by the two world
wars and the Great Depression of the 1930s.
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Piecewise Defined Functions
Sometimes we need more than a single formula to
specify a function algebraically. In this case
the formula used to evaluate the function depends
on the value of x.
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Piecewise Defined Functions
The following is a quick example of a piecewise
defined function
Notice
26.5
53.8
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Piecewise Defined Functions
The following is a quick example of a piecewise
defined function
Notice that the domain of f , in this case, is
the set all real numbers. That is, Dom f (? ,
?)
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Piecewise Defined Functions
The percentage p (t) of buyers of new cars who
used the Internet for research or purchase since
1997 is given by the following function. (t 0
represents 1997).
Notice that the domain of p is the interval 0 ,
4. That is, Dom p 0 , 4. The model is
based on data through 2000. Source J.D. Power
Associates/The New York Times, January 25, 2000,
p. C1
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Piecewise Defined Functions
This notation tells us that we use the first
formula, 10t 15, if 0 ? t lt 1, or, t is in
0, 1) the second formula, 15t 10, if 1 ? t ?
4, or, t is in 1,4
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Piecewise Defined Functions
Thus, for instance, p(0.5) 10(0.5) 15
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0.5 lt 1, or, equivalently, 0.5 is in 0,
1). p(2) 15(2) 10 40 We used the second
formula since 1 ? 2 ? 4, or equivalently,
2 is in 1, 4. p(4.1) is undefined p (t
) is only defined if 0 ? t ? 4.
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Graphically Specified Function
The most common method for visualizing a function
is its graph.
The graph of a function is the set of all points
(x, f (x)) in the xy-plane such that x is in the
domain of f .
Sometimes the function is only known through its
graph and may be very difficult to represent it
algebraically. The next example illustrates this
case.
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Graphically Specified Function
The vertical acceleration a of the ground as
measured by a seismograph during an earthquake is
a function of the elapsed time t. The figure
shows a graph generated by seismic activity
during the Northridge earthquake that shook Los
Angeles in 1994. For a given value of t,
the
graph provides a
corresponding value of a.
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Graphically Specified Function
Example The monthly revenue R from users logging
on to your gaming site depends on the monthly
access fee p you charge according to the formula
(R and p are in dollars.) Sketch the graph of R.
Find the access fee that will result in the
largest monthly revenue.
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Graphically Specified Function
Solution To sketch the graph of R by hand, we
plot points of the form (p , R(p)) for several
values of p in the domain 0 , 2.5 of R. First,
we calculate several points.
p 0 0.5 1 1.5 2 2.5
R(p) 0 5600 8400 8400 5600 0
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Graphically Specified Function
Graphing these points gives the graph in the
figure on the left, suggesting the parabola shown
on the right.
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Graphically Specified Function
The revenue graph appears to reach its highest
point when p 1.25, so setting the access fee at
1.25 appears to result in the largest monthly
revenue.
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Graphically Specified Function
Example The following table gives the weights
(in pounds) of a particular child at various ages
(in months) in her first year.
Age t 0 2 3 4 5 6 9 12
Weight W 8 9 13 14 16 17 18 19
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If we represent the data given in the table
graphically by plotting the given pairs (t
,W(t)), we get, (we have connected successive
points by line segments)
W(5)
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W(4.5) ?
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Graphically Specified Function
Example Given the graph of y f (x), find f (1).
f (1) 2
(1, 2)
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Graphically Specified Function
Using the definition of graph of a function and
the calculations done in the previous examples we
can now see how to determine the domain and range
of a graphically defined function.
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Example Determine the domain, range, and
intercepts of the function defined by the
following graph.
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Graph of a Function
Vertical Line Test The graph of a function can
be crossed at most once by any vertical line.
Function
Not a Function
It is crossed more than once.
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y
x
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y
x
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Sketching a Piecewise Function
Sketch the portion of the formula on its domain
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Useful Functions and Their Graphs
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