Chapter 1
Functions and
Linear Models
Sections 1.1 and 1.2
Functions and Linear Models
 Functions: Numerical, Algebraic
 Functions: Graphical
 Linear Functions
 Linear Models
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.
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.
A way to picture a function is by an
arrow diagram
f
x
y
x
y
x
X
DOMAIN
Y
Not in the
range of f
RANGE
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
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.
Algebraically Defined Function
Is a function represented by a formula. It has
the format y = f (x) = “expression in x”
Example: f ( x)  3x  2 is a function.
2
f (5)  3(5)2  2  77
Substitute 5 for x
f ( x  h)  3  x  h   2
Substitute x+h for x
2
 3x  6 xh  3h  2
2
2
Algebraically Defined Function
Is a function represented by a formula. It has
the format y = f (x) = “expression in x”
Example: f ( x)  3x  2 is a function.
2
In this case the natural domain of the function
is the set of all real numbers. That is,
Dom f = (– , )
Algebraically Defined Function
4
is a function.
Example: s(t ) 
t 1
In this case the natural domain of the function
is the set
Dom s  t | t  1  0  t | t  1
In interval notation this is
Dom s  ( ,1)  (1,  )
Algebraically Defined Function
Example: h( z )  2  3z is a function.
In this case the natural domain of the function
consists of all values of z such that
2  3z  0 or 3z  2 or z  2/ 3
In interval notation this is
Dom h  ( 2 / 3, )
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
y = f (x) f (x1)
x2
…
…
xn
f (x2)
…
…
f (xn)
Numerically Specified Function
Example: Suppose that the function f is specified by the
following table.
x
f (x)
0
1
2
3.01 -1.03 2.22
3.7
4
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
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,
P(1950)  2,560, 000, 000
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.
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
Numerically Specified Function
V(t) ft/sec
8
7
6
5
4
3
2
1
-1
1
-1
2
3
4
5
6
7
t in seconds
8
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.
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
12 t
r 
 r 
P  1  
12  12 

M
12 t
r 

1    1
 12 
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?
A( x )   x  500 x
2
Dom A  x | 0  x  500  (0,500)
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.
Mathematical Modeling
The pattern of the data points suggests exponential
growth.
Mathematical Modeling
We use a graphing calculator with exponential regression
capability to apply the method of least squares and obtain
the exponential model p  (0.008079266)  (1.013731)t
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.
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.
Use when x values
expression 1 if condition 1 satisfy condition 1
expression 2 if condition 2

f ( x)  

expression n if condition n Use when x values
satisfy condition n
Piecewise Defined Functions
The following is a quick example of a piecewise
defined function
Use when x values are
less than or equal to 2
32  5.5 x if x  2
f ( x)  
2
13.8  2.5 x if x  2
Use when x values are
Notice
greater than 2
f (1)  32  5.5(1) = 26.5
f (4)  13.8  2.5(4) = 53.8
2
Piecewise Defined Functions
The following is a quick example of a piecewise
defined function
32  5.5 x if x  2
f ( x)  
2
13.8  2.5 x if x  2
Notice that the domain of f , in this case, is the
set all real numbers. That is, Dom f = (– , )
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).
10t  15 if 0  t  1
p(t )  
15t  10 if 1  t  4
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
Piecewise Defined Functions
10t  15 if 0  t  1
p(t )  
15t  10 if 1  t  4
This notation tells us that we use
the first formula, 10t + 15, if 0  t < 1, or, t is in [0, 1)
the second formula, 15t + 10, if 1  t  4, or, t is in [1,4]
Piecewise Defined Functions
10t  15 if 0  t  1
p(t )  
15t  10 if 1  t  4
Thus, for instance,
p(0.5) = 10(0.5) + 15 = 20
p(2) = 15(2) + 10 = 40
Here we used the first formula since
0  0.5 < 1, or, equivalently, 0.5
is in [0, 1).
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.
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.
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.
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( p )  5600 p 2  14000 p
0  p  2.5
(R and p are in dollars.) Sketch the graph of R. Find the
access fee that will result in the largest monthly revenue.
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
R(p)
0
0.5
1
1.5
2
5600 8400 8400 5600
R( p )  5600 p  14000 p
2
2.5
0
0  p  2.5
Graphically Specified Function
Graphing these points gives the graph in the figure on
the left, suggesting the parabola shown on the right.
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.
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
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) = 16
W(4.5) 
Graphically Specified Function
Example: Given the graph of y = f (x), find f (1).
f (1) = 2
(1, 2)
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.
Example: Determine the domain, range, and intercepts
of the function defined by the following graph.
y
4
(2, 3)
(10,0)
0
-4
(1, 0)
(0, -3)
(4, 0)
x
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.
y
x
Not a function
y
x
A function
Sketching a Piecewise Function
2  x if  2  x  1
f ( x)   2
 x +1 if 1  x  2
Sketch the portion
of the formula on
its domain
Useful Functions
and Their Graphs