June, 2006: This page now requires Internet Explorer 6+MathPlayer or
Mozilla/Firefox/Netscape 7+.
2. Functions Introduction and Review.
Preface: There are many
different kinds of numbers and ways to express, compare, and relate them.
Wherever we turn, whatever we read, we find numbers used to describe situations
while the relations between numbers help us understand, plan, and control
these situations. The special kind of relation between numbers that
is called a function, first recognized explicitly by Leibniz, evolved
through the work of the Swiss mathematician Leonhard Euler (1707-1783)
( pronounced "Oiler") and the French mathematician Joseph Lagrange (1763-1813)
to become a cornerstone for mathematics in the 20th century. Though Newton,
Leibniz and the other early developers of the calculus did not have the
current concept of a function to assist them, we would handicap ourselves
seriously if we did not recognize and use the function concept today.
Most
students in a precalculus course have had some experience with functions in
their previous studies. A complete and detailed introduction and review of
function concepts and examples early in this course would delay our reaching
the object of our studies, the analysis of the functional relations that are
the background for a calculus course. For this reason, our review
in this section will be limited to an exploration of the basic nature of
functions and ways to think about functions that make sense for describing,
visualizing and analyzing them. Thus our attention is directed primarily
at functions and change. Throughout this course we will develop a catalog
of definitions and fundamental properties of functions used in the further
study of mathematics, especially in the calculus.
The Trip: One simple and very useful context for understanding
functions is that of a car traveling on a road, what we might call "the trip."
Numbers that are interesting here are often found on the dashboard of the
car and by the roadside. They measure such quantities as the amount of gas
in the gas tank, the distance traveled during this trip, the distance traveled
by the car since it was "new," the temperature
of the engine, the mileage markers on the side of the road, the speed limit
markers, the speedometer giving the car's speed, the tachometer indicating
the rate at which the engine is turning, and the clock.
For
sure when we are considering any trip in a car we are aware that time is changing
whether the car is standing still or moving very quickly. Thinking about the
relations between these quantities can help us gain some informal understanding
of functions and how they can be used to understand the trip. For example,
the distance the car has traveled is related to the gas the car's engine
has consumed. The temperature of the engine is related to the rate at which
the engine is turning. All the measurements can be thought of as related
to time, since when we make a measurement we do that at a particular time.
The concept of function can be applied in
many contexts beyond that of real numbers. For the purpose of beginning
our study of functions, we will use the concept only for examining functions
that relate real numbers. These functions are usually described as real
valued functions of a real variable. Other function concepts studied
in mathematics treat relations between three or more variable numerical
quantities as well as variables that are not numbers but geometric points
and vectors.
Static Functions as Relations:
The mathematical concept of function connects quantities while suggesting
informally something stronger than just a relation between the quantities.
In one sense when we describe a function relationship between variable
quantities, we are establishing a priority or an order to the information.
When two variables are related we often can identify one as being an independent,
governing, or controlling variable while describing the other as a
dependent, regulated, or controlled variable. These descriptives indicate
an important quality of a function, namely that knowledge, assignment,
or specification of one variable's value will determine the value of the other
variable. In this sense we can consider each of the variables in
the trip context as functions controlled by the time variable. It is not that
there is any necessary scientific or causal relation between time and the
position of the car (or any of the car variables). The function language merely
indicates that knowing the time should allow us to determine the position
of the car.
A function connects the information about the variables by pairing the data
and assigning a priority to the pairs.Knowing the first number of
a pair uniquely determines the pair, and thus the second number of the pair.
Dynamic Functions -
paired changes. There is a second way to
think of functions relating variables. We consider the function as
a mechanism or interpreter that transforms one measurement or number
into a second number. [It helps to avoid thinking of this process
as being strictly causal in physical situations, even though there is
some strong connection to causality with a priority to the order of
the relation.]
In the trip context we can think of the car dashboard as a mechanism.
Knowing the time by reading the clock allows us to determine what the
reading on the odometer is. The distance our car has traveled is a
function of time. One can also connect the amount of gas in the
gas tank, measured by the gas gauge, to the distance the car has traveled
during a trip, thus the amount of gas is a function of the distance
traveled.
The history of clocks and watches and the
measurement of time are a fascinating part of the development of science
and mathematics. Time certainly plays an important role in our lives
today, one that we sometimes overlook too easily. We take for granted
our ability to measure time with precision to the microsecond, but in
1773 the British government paid a 20,000 pound sterling prize to John
Harrison for developing a very accurate navigation chronometer with
which to calculate longitude.
This mechanistic view of functions goes particularly well
with the use of calculators. You enter a number on the display, push
a single button and the (resulting) number on the display is usually changed.
The terms "input" and "output" are used here to
describe the values of the controlling and the controlled variables respectively.
When thought of as a machine, a function will process some numbers while
being unable to process others. For example, if the process returns the multiplicative
inverse (reciprocal) of a number, then the process will work well on all numbers
with the single exception of 0. Or if the process returns the real number
square root of a given number, then the process will operate on non negative
real numbers but will fail to return a result for negative real numbers. [Remember,
the square root of a negative number is not a real number, but an imaginary
number.]
Figure illustrating a function machine.
We
refer to the collection of numbers which the function can process as
the source or domain of the function. Thus the function that returns
the reciprocal of a number has a source of the set of all non zero real numbers,
while the source for the real valued square root function is the set of non negative
real numbers. The source of a function when not described explicitly is assumed
to contain all the inputs that work.
It is useful to describe the source of a function so you can avoid errors
that sometimes arise from applying the function in meaningless situations.
The most common algebraic restrictions on the source of a function arise from
division by 0 and finding square roots of negative numbers.
For example,
the function which returns the number `1/{1-x^2}` for an input of the number x can have a source that includes all real
numbers except 1 and -1.
As a second
example, the function that returns the number `sqrt{1-x^2}`
for the input number x can have a source that includes only numbers
in the interval [-1,1].
Notation
for Describing Functions: As these last examples demonstrate, it would
be convenient to have a notation that allows us to describe a function either
from the static view of a collection of ordered pairs or from the dynamic
view by designating what the function yields as output for a specified, yet
arbitrary, input. The notation and vocabulary that has evolved (and is still
evolving) must distinguish at least three things: the first number of
the pair, called the argument or the input number, the second number of the
pair, called the result, value, or the output number, and the designation
or name for the function.
Variable
Relations: A traditional approach to this
notation considers 2 variable names, say x and y, with
x representing the first number for the function and y
representing the second number. In this approach there is no name
for the function, only a statement that y is a function of
x and an equation describing an explicit relation between the
variables, such as y = 2x+3, or an implicit relation
such as `x = y^2` and `y>=0`. This approach
abuses the notation since the letter y refers both
to the number and the function relation of that number to the number
x.
For example, when the function y is described by the equation
y = 2x+3 we can determine the value of y
when x = 6 by substituting 6 for x in the formula 2x+3
to find y = 2 (6)+3=15.
The ordered pairs determined by this function have
the form `(x,y)` where `y=2x+3` or `(x,2x+3)`. For `x = 6`, the information
from the function is that (6,15) is one of the pairs of the function.
The evaluation of y when x=6 has a traditional notation
of
`y| _{x=6}=2x+3| _{x=6}=2(6)+3=15`
Beware: A common error is confuse the
parentheses in this notation for the value of a function for
the parentheses used frequently to collect terms in an expression
to be treated as a single number in some algebraic calculation like
multiplication.
By convention, when f is a function, the symbols f(t+5)
means the value of f for the number described by t+5,
it does not mean the product of the number f by the number
(t+5).
Function Values: A more contemporary
approach to the notation (based on the notation used by Lagrange in his work
at the end of the 18th century), with which you are no doubt somewhat familiar,
assigns to the function a specific letter or symbol that suggests what
the function does or at least gives the function a recognizable name.
For example, the function might be the square root function. The square
root function is assigned the shorthand name sqr, or, as with less
familiar functions, it might be named temporarily for the purpose of discussion
with the symbol f or g. Once the name has been given, the relation
between input and output, argument and result, can be described in a number
of ways. A variable name is given, or presumed, for the input, like x
or t, and then the output is described as the value of the function
at x or t, which is denoted most often as f(x)
or f(t). Notice that this last notation for the value
of the function at x has four symbols: f denotes the name of
the function, x denotes the name of the number to be input, the two
parentheses separate the name of the function from the name of the variable.
A description of the function can then be accomplished by giving a
procedure for finding the value of f(x), such as f(x)=2x+3.
Once `f` has been described and thus defined for the purpose of
discussion, the notation allows us to denote the value of the function
that corresponds to the number 6 as `f(6)`. This number can be computed
using the defining equation: `f(6)=2(6)+3=9`. Likewise the
value of the function for the number `-4` is denoted `f(-4)`, and can be
computed to be `f(-4)=2(-4)+3=-5`. For the expression `t+5`,
which can represent a number, the value of the function is denoted `f(t+5)`,
which can be simplified by using the definition of `f` so that `f(t+5)=2(t+5)+3
= 2t+13`.
The function
value notation is sometimes combined with the two variable notation in statements
such as "suppose `y` is a function of `x`, with `y=f(x)=2x+3`." Then evaluation can be expressed by the equation: `y| _{x=6}=f(6)=2(6)+3=15`.
Transformation Notation: Another
current approach to describing the input/output information of a function
is to give the name of the function, then a name for an input variable followed
by an arrow and an expression describing the output that results from the
input. For example, `f : x -> 2x+3`
or more generally `f : x -> y
= f(x)`.
The arrow in the notation helps convey visually the dynamic aspect of a function
that transforms a number. This notation helps underscore the active nature
of those functions that in some way do require a construction of the resulting
number values by some conceptual (and perhaps even mechanical or electronic)
process. Multi-case functions: In
some contexts a function cannot be described by a single simple algebraic
formula using well known conventional functions. This can be for many reasons:
There
may be no algebraic formula that captures that the relation;
The function
may arise from a non algebraic context;
The function
may piece together some more simple function.
To piece together simple functions, the most
common method is to establish some tests on the source numbers. These tests
determine precisely what the appropriate method is for determining the function's
value.
Here is an example of a function F defined by cases.
Case 1: if `x <= 0`, then `F(x)
= 0`;
Case 2: if `0< x <4` then `F(x)
= {x^2}/16, and
Case 3: if `4 <= x` then `F(x)=1`.
To
use this definition of `F` we need to first check under which case the
argument for evaluation falls, then follow the function's rule as appropriate.
So, to
find `F(-3)` we see first that `-3` falls under Case 1, so `F(-3)
= 0`.
To
find `F(3)` we check that `3` is under Case 2, so `F(3) = {3^2}/16
= 9/16`.
Finally
to find `F(5)` we use Case 3, so `F(5) = 1`.
The conventional notation for this function's definition is expressed by
a multi-lined equation:
Functions
and Tables: In some cases we do not know
any formula that precisely matches the pairings for a function. This
is in fact mot common when we look at any function that arises
from real world phenomena. We may have a formula that provides
a good estimate or we have only some data . Sometimes this data is
measured and recorded at selected numbers (or times). In this case
the information we have for the function may be displayed in the form
of a table.
For example if f is the function
recording the room temperature we may know only that f(0)
= 60, f(4) = 55, f(8) = 65, f(12) = 68, f(16)
= 68, f(20) = 68, f(24) = 60. It is convenient to
put this data in a table. See Table 1.
Be careful not to assume too much about a function that is represented
only by a table of values at selected numbers. For example, although
f(0)= 60 and f(4)=55, you cannot assume that f(2)
is between 60 and 55.
x
f(x)
0
60
4
55
8
65
12
68
16
68
20
68
24
60
Table 1
Non algebraic [Geometric] Functions: If we think of the room temperature
function again, we can imagine a device recording the temperature graphically
with a sheet of paper moving at a steady rate under a marking pen attached
to a temperature sensitive device that moves the pen depending on the room
temperature. This would give a rather different record of the room's temperature
as a function of time. Once scales are established on the paper for the time
and temperature, we can give a better estimate of the temperature at many
different times by being able to read the graph. This graphical presentation
of a function gives much more information than a table, but you should be
careful here as well not to infer too much from the graph, especially since
the mechanism, the recording instrument, and the scales all can contribute
to the imprecision of the information.
Curves and
coordinates in Cartesian geometry. Using numbers to study figures in geometry
is not a very recent part of mathematics. And using figures to study connections
between numbers is also not new to math. In your previous course work you have
studied the algebraic relations between coordinates of points on lines, circles,
and other curves and conversely have graphed figures to illustrate the relation
between numbers involved in an equation. The key in these correspondences
has been the connection usually attributed to Rene Descartes in analyzing
curves with numbers through use of common measurements and variables related
by equations determined by the geometry, called analytic (or Cartesian) geometry.
Coordinate system analysis applied to planar curves can lead to function
relations that are commonly encountered in science. Measuring devices record
the changes during some experiment as a curve. The variables connected to
the curve are associated with the experiment using rectangular coordinates
and then the curve is interpreted as representing the relation between these
variables. Without any prior knowledge of the relation of the variables,
neither can be assumed to be controlling, but there is a general convention
to consider the horizontal, first coordinate variable (X) as controlling
with the vertical and second coordinate (Y) as the controlled variable.
But for a curve to give a function relation between these variable there
needs to be more either understood implicitly or made explicit.
In particular, when given a value for X we need a way to determine a value
for the Y variable using the curve as the mechanism for that determination.
This is easy enough in many familiar cases. For the given value of X, say
a, find a point P on the curve with that value as its first coordinate,
i.e., P has coordinates (a,b) for some number b. When there
is only one point P with first coordinate a on the curve, then the
value b is uniquely determined by the curve, and b is the value
of the Y variable corresponding to a. In this case we say that the
curve has determined Y as an explicit function of X and assume
we are using the coordinate convention just described.
The Slope
of the Tangent to a Curve as a Function: There are other functions we
can associate with curves in analytic geometry. For many curves we can determine
at each point on the curve a line that very close to the point looks indistinguishable
from the curve and yet close to the point meets the curve only once. As mentioned
in section 0.A, these lines are sometimes referred to as touching or tangent
to the curve at the point, or tangent lines. To repeat the example from 0.A,
to find the tangent line to a circle at a point P, you need only draw a radius
from the center of the circle to the point P and then construct the line perpendicular
to the radius at P, which by Euclidean geometry must be the tangent line.
So how does this give rise to a function using numbers?
Consider the case when the curve determines Y as an explicit function of X.
For a given value of X, say a, we again find a unique point P
on the curve with that value as its first coordinate, i.e., P has coordinates
(a,b) for a unique number b. Now it sometimes turns out that
we can find a unique, non vertical tangent line to the curve at P and determine
the slope of this line, which we will call m. Since the value of
m is uniquely determined using the curve from the value of X, m
is a function of X. The value of m is derived geometrically from the
original curve using the measurement of the slope of the tangent line at the
point P determined by a.
Area of Geometric
Figures as Functions: One of the most frequently encountered problems
in geometry is that of finding a general method for determining the area
enclosed by a class of planar figures. You have learned formulae for areas
of squares, rectangles, triangles, trapezoids, circles, and perhaps two or
three other general shapes. The measurement of these areas is usually based
on other measured features of the figures, such as lengths of sides or relevant
line segments and sometimes even the size of angles.
The relationships between area and these other variables of geometric figures
can often be described as functions. For example we can consider rectangles
that have a base of length 20 centimeters and determine the area when the
altitude has length l centimeters. Not too hard.
The area is 20 l square centimeters. Or in the same setup, we can determine
the length of the altitude when the rectangle has area A square centimeters
with almost as little effort to be `A/20` centimeters. It requires a
little more thought to determine the length of the diagonal of the rectangle
with this setup based on the area A to be `sqrt{A^2+1600}/20`.
Another familiar
area relation is found in the circle where the equation `A=pir^2` allows
us to determine either the area or the radius of a circle by knowing the other
measurement. A slightly more subtle area relation was described in section
0.A. As the example there demonstrated there are many ways to measure a region
in the plane, like a triangle, and passing a line across that region can give
an area function determined by the position of the line.
Implicit
functions: Let's consider the equation 3X + 4Y = 24 where X and
Y represent real numbers. There are many possible choices for X and
Y, some of which will make the equation true (say X = 4 and
Y = 3 ) and some of which will not (say X = 1 and Y = 2). Given a
value for X there is one and only one value for Y which will
make the equation true. Thus if X = 1 then for the equation to be
true we have 3 + 4Y = 24 , so 4Y = 21 and Y = 5.25. Since the equation
determines a unique value for Y from any choice of X, we can say that
the equation has determined Y implicitly as a function of X.
A more
subtle relation is presented by the equation X^{2} +
Y^{2} = 25. This equation can be satisfied by many functions.
For example `f(x)=sqrt{25-x^2}`,
`g(x)=-sqrt(25-x^2)`,
or .
Here the equation does not determine a single value of one variable
from the choice of another value but allows many possible functions
which will satisfy the equation when the Y is determined by one of
these functions. In this case the functions are described as
being defined implicitly by the equation.
Comment: With only a limited list of function
values there is no way to tell what the function might have
for its graph without assuming some more restrictive qualities.
For example, suppose the graph of the function is known
to pass through two points with coordinates (0,0) and (1,1).
You might think this is the function with `f(x) = x`, but in fact
it could be `f(x)=x^2` or `f(x) = x^3`
or
`f(x) = x^2 + x - x^3`.
In fact if we consider all functions with f(0)= 0
and separately those functions with f(1)=1, we are only asking
for those functions that satisfy both conditions, which doesn't
seem like a large restriction when we think of all the different
ways we could connect the points with coordinates (0,0) and (1,1)
in a Cartesian plane.
Visualizing
Functions and Transformation Figures: The key idea
in visualizing functions with mapping diagrams or transformation figures is
to have two parallel number lines representing the source (domain) and
the target (range). The function can be thought of as a process relating
points (numbers) on these two lines.
A point element on the source line is
chosen which corresponds to a number. The function is applied to that number,
and the resulting value is found represented on the target line. An arrow
drawn from the point on the source line to the corresponding point on the
target line visualizes the relation between the corresponding numbers.
In
one sense, the transformation figure is a visualization of a function
table. The numbers in the two columns of the table are represented
by points on the two lines in the figure. The function relation
that the table displays implicitly by having corresponding numbers
in the same row is visualized in the figure by the arrow.
While the relative size of the numbers in the target column of the
table is not represented in the display, the transformation figure
uses the number line order to represent this aspect of the function's
values.
Here is an illustration that should help you see
some of these features. You can work on other examples after this
one to begin to see some of the power of this visualization. [ To
see a dynamic example of a transformation figure for linear functions,
follow
this link.]
Example: Suppose f(x)=2x+
3. Table 1 shows a selection of the values this function relates, while
this same information is visualized in Figure 4. Notice that larger
numbers in the source column of the table correspond to larger values
in the target column. On the transformation figure this feature can
be seen by the fact that the lines connecting the corresponding points
on the source and target lines do not cross. This is evidence of a function
with increasing values.
x
f(x)=2x+3
5
13
4
11
3
9
2
7
1
5
0
3
-1
1
-2
-1
-3
-3
-4
-5
-5
-7
Table 1
Figure 4
Graphs
of Functions and Other Relations: In your
previous work with functions and equations you have worked extensively with
the graphical visualization using Cartesian coordinates for the plane to identify
the function pairing of numbers.
In
the graph of a function f we identify the pair of numbers a
and f(a) with the point in the plane with coordinates (a,f(a)).
We can plot marks at many of these points but when the domain of the function
is an interval or as is more common all real numbers, we cannot hope to plot
all the points. Instead we try to give a sense of how the points are related
by drawing a curve that passes through some points that are known to be on
the graph of the function. In doing this we are drawing figures much as students
in elementary school draw figures by connecting the dots in order, or as economists
graph the hour to hour price of some stock on the stock market or as a chemist
would visualize the minute by minute temperature reading on a laboratory thermometer
during an experiment.
Here
are some examples of transformation figures and graphs. On the left are the
tables of values for the functions at selected points, while on the right
are the corresponding figures and graphs. [Graphs and Figures made using Winplot.]
Example
0.B.2 f(x) = x
x
f(x)=x
2
2
1
1
0
0
-1
-1
-2
-2
Example
0.B.3 f(x) = -x
x
f(x)=-x
2
-2
1
-1
0
0
-1
1
-2
2
Example
0.B.4 f(x) = |x|
x
f(x)=|x|
2
2
1
1
0
0
-1
1
-2
2
Example
0.B.5 f(x) = 2x
x
f(x)=2x
2
4
1
2
0
0
-1
-2
-2
-4
Example
0.B.6 f(x) = x+1
x
f(x)=x+1
2
3
1
2
0
1
-1
0
-2
-1
Example
0.B.7 f(x) = x^{ 2}
x
f(x)=x^{ 2}
2
4
1
1
0
0
-1
1
-2
4
Example
0.B.8 f(x) = 1/x
x
f(x)=1/x
2
0.5
1
1
0
??
-1
-1
-2
-0.5
Example
0.B.9 f(x) = 2^{x}
x
f(x)=2^{x}
2
4
1
2
0
1
-1
0.5
-2
0.25
Example
0.B.10 f(x) = 3
x
f(x)=3
2
3
1
3
0
3
-1
3
-2
3
Example
0.B.11 f(x) =-2x + 1
x
f(x)=-2x+1
2
-3
1
-1
0
1
-1
3
-2
5
Exercises: The exercises for this section cover material
which you may recall from previous course work in algebra. You may not recall
all of these topics or how to do the problems precisely. You may want to refer
to the texts or notes from your previous courses in mathematics if you find
these difficult. The skills needed to solve these problems will be important
in the work ahead- so be careful to identify any difficulties you have with
these problems and try to remedy any misunderstandings as you proceed. 1. For this problem let f be defined by f (x) = 5x^{
2} + 3. a) Find the following. Simplify you answer when possible.
i) f(1)
iii) f(1+h)
ii) f(h)
iv) [f(1+h) - f(1)]/h
b) Find any
number(s) z where f(z) = 23. c) For which values of x is
f(x) < 23? Express your answer as an interval.
2. USING INTERVAL
NOTATION, express the largest set of real numbers that can serve as the
domain of each of the following functions:
a)
f(x) = (4 - x^{ 2})/(x + 2)
b) g(x) = 1/[(4-x^{ 2})]
3. Suppose that F is defined by .
Sketch a transformation figure and a complete graph of f.
Determine the domain and the range of f.
4. Solve for x:
a)
3^{ x-2} = 3^{ 7-2x} b) 4^{
3x} = 8 c) 1/_{3}(x
- 5) = 2 d) 1/30 - 1/x = 1/6
5. Boyle's law states that, for a certain gas P*V = 320, where P is
pressure and V is volume.
(a) Draw a complete
graph representing this situation. Label your axes and write an equation for
each asymptote.
(b) If `8 <= V <= 40`, what are the corresponding values of
P?
6. Let f(x) = x^{ 2} + 4x
- 5. A. Find the axis of symmetry and the vertex of f. B. Sketch
a graph of f labeling clearly the coordinates of the vertex and the
X- and Y- intercepts.
7. Old McDonald has a farm ,and on that farm she has some sheep and a
pasture with a 200 meter long stone wall. She wants to enclose a rectangular
section of the pasture for a small sheep pen using the wall for one side and
140 meters of fencing she was given by her uncle Milo for the other three
sides.
A. Let x denote the length of the fence that will be attached to the
wall used as a side for the pen. Which of the following equations express
the area of the pen, A, as a function of x?
a. A = x ( 70 - x)
b. A = 2 x ( 140 - x)
c. A = x ( 140 - x)
d. A = x ( 200 - (1/2)x)
d. A = 2 x ( 70 - x)
e. A = 2 x ( 200 - x)
17. Write a short story about cooking dinner. Discuss briefly
those aspects of the meal's preparation that might be related by functions.
18. Write a short story about going shopping. Discuss briefly
those aspects of the shopping that might be related by functions.
19. Write a short description of an ecological system.
Discuss briefly those aspects of the system that might be related by functions.
20. Write a short description of the human body. Discuss
briefly those aspects of the body that might be related by functions.
21. Write a short story about an athletic event or sports
competition. Discuss briefly those aspects of the story that might be related
by functions.