Transcript X - Columbus State University

Sections 3.1 and 3.2
Relations and Functions
A Refresher on Set Theory
Experience says that a brief refresher on some
basic notions is welcome, if not completely
necessary, at this stage. To that end, we present a
brief summary of ‘set theory’ and some of the
associated vocabulary and notations we use in the
text. Like all good Math books, we begin with a
definition.
A Refresher on Set Theory
Definition: A set is a well-defined collection of
objects which are called the ‘elements’ of the set.
Here, ‘well-defined’ means that it is possible to
determine if something belongs to the collection or
not, without prejudice.
A Refresher on Set Theory
For example, the collection of letters that make up
the word “smolko” is well-defined and is a set, but
the collection of the worst math teachers in the
world is not well-defined, and so is not a set.
In general, there are three ways to describe sets.
They are,
Ways to Describe Sets
1. The Verbal Method: Use a sentence to define a
set.
2. The Roster Method: Begin with a left brace { ,
list each element of the set only once and then
end with a right brace }.
3. The Set-Builder Method: A combination of the
verbal and roster methods using a ‘dummy
variable’ such as x.
Ways to Describe Sets
For example, let S be the set described verbally as
the set of letters that make up the word ‘smolko’.
A roster description of S would be {s, m, o, l, k}.
Note that we listed ‘o’ only once, even though it
appears twice in ‘smolko’.
Also, the order of the elements does not matter, so
{k, l, m, o, s} is also a roster description of S.
Ways to Describe Sets
A set-builder description of S is:
{x | x is a letter in the word ‘smolko’}
The way to read this is: The set of all elements x
such that x is a letter in the word ‘smolko’.
In each of the above cases, we may use the familiar
equals sign  and write S  {s, m, o, l, k} or
S {x | x is a letter in the word ‘smolko’}
Ways to Describe Sets
S {x | x is a letter in the word ‘smolko’ }
{s, m, o, l, k}
Clearly m is in S and q is not in S. We express
these sentiments mathematically by writing m  S
and q  S.
Sets of Numbers
1. The Empty Set: {}{x| x  x}. This is the set
with no elements. Like the number ‘0’, it plays
a vital role in mathematics.
2. The Natural Numbers:  {1, 2, 3, …}.
3. The Integers:  {…, 3, 2, 1, 0, 1, 2, 3, …}.
Sets of Numbers
4. The Rational Numbers:
 {a/b | a   and b   and b  0}.
It turns out that another way to describe the
rational numbers is:
 {x | x possesses a repeating or terminating
decimal representation}.
Sets of Numbers
5. The Real Numbers:
 {x | x possesses a decimal representation}.
6. The Irrational Numbers:
 {x | x is a non-rational real number}.
Said another way, an irrational number is a
number whose decimal representation neither
repeats nor terminates.
Sets of Numbers
7. The Complex Numbers:
 {x | x  a+bi and a,b   and i  1 }.
Despite their importance, the complex numbers
play only a minor role in the text.
Interval Notation
For the most part, this course focuses on sets
whose elements come from the real numbers .
Recall that we may visualize  as a line.
Segments of this line are called intervals of
numbers.
Below is a summary of the so-called interval
notation associated with given sets of numbers.
Interval Notation
Examples
As an example, consider the sets of real numbers
described below.
Intersection and Union of Sets
We will often have occasion to combine sets. There
are two basic ways to combine sets: intersection
and union. We define both of these concepts below.
Definition: Suppose A and B are sets.
 The intersection of A and B is defined to be the set
A B {x | x  A and x  B }.
 The union of A and B is defined to be the set
A B {x | x  A or x  B or both}.
Intersection and Union of Sets
Examples: If A  [5,3) and B  (1,), then we
can find A B and A B graphically.
To find A B , we shade the overlap of the two and
obtain A B  (1,3).
To find A B, we shade each of A and B and
describe the resulting shaded region to find
A B  [5,).
Intersection and Union of Sets
If A  [5,3) and B  (1,), then we can find
A B and A B graphically.
More Examples
Express the following sets using interval notation.
1. {x | x  2 or x  2}
2. {x | x  3}
3. {x | x   3}
4. {x | 1< x  3 or x  5}
Solutions
1. {x | x  2 or x  2}
Solutions
2. {x | x  3}
Solutions
3. {x | x   3}
Solutions
4. {x | 1< x  3 or x  5}
The Cartesian Coordinate Plane
In order to visualize the pure excitement that is
Pre-calculus, we need to unite Algebra and
Geometry. Simply put, we must find a way to draw
algebraic things. Let us start with possibly the
greatest mathematical achievement of all time:
The Cartesian Coordinate Plane.
The Cartesian Coordinate Plane
Imagine two real number lines crossing at a right
angle at 0 as drawn below.
The Cartesian Coordinate Plane
The horizontal number line is usually called the xaxis while the vertical number line is usually
called the y-axis.
As with the usual number line, we imagine these
axes extending o indefinitely in both directions.
The Cartesian Coordinate Plane
Having two number lines allows us to locate the
positions of points off of the number lines as well
as points on the lines themselves.
For example, consider the point P on the next slide.
To use the numbers on the axes to label this point,
we imagine dropping a vertical line from the x-axis
to P and extending a horizontal line from the y-axis
to P.
The Cartesian Coordinate Plane
This process is sometimes called ‘projecting’ the
point P onto the x- (respectively y-) axis.
The Cartesian Coordinate Plane
The projections of P onto the x and y axis are called
respectively, the x and the y coordinates of P.
The Cartesian Coordinate Plane
This way P is represented by the ordered pair of
numbers x  2 and y  4. We write P  (2,4).
The Cartesian Coordinate Plane
In general, any point in the plane is an ordered pair
on real numbers. P  (x,y).
P  (x , y)
The Cartesian Coordinate Plane
Therefore, the plane itself is the set of all possible
pairs P  (x,y) of real numbers.
P  (x , y)
The Cartesian Coordinate Plane
The axes divide the plane into four regions called
quadrants. They are labeled with Roman numerals
and proceed counterclockwise around the plane:
Relations and Functions
Relations
In certain sense, all of Pre-calculus can be thought
of as studying sets of points in the plane.
With the Cartesian Plane now fresh in our memory
we can discuss those sets in more detail and as
usual, we begin with a definition.
Definition: A relation is a set of points, (ordered
pairs) in the plane.
Relations
Since relations are sets, we can describe them using
the techniques presented before. That is, we can
describe a relation verbally, using the roster
method, or using set-builder notation. Since the
elements in a relation are points in the plane, we
often try to describe the relation graphically or
algebraically as well.
Relations
Depending on the situation, one method may be
easier or more convenient to use than another.
As an example, consider the relation
R {(2,1), (4,3), (0,3)}
As written, R is described using the roster method.
Since R consists of points in the plane, we follow
our instinct and plot the points.
Relations
Doing so produces the graph of R.
Examples of Relations
Graph the following relations.
Solutions
The graph for examples 1 and 2 are,
Solutions
The graph for example 3 is,
Solutions
The graph for example 4 is,
Solutions
The graph for example 5 is,
Solutions
The graph for example 5 is,
Relations defined by Equations
In this section, we delve more deeply into the
connection between Algebra and Geometry by
focusing on graphing relations described by
equations. The main idea of this section is the
following.
Relations defined by Equations
The graph of an equation is the set of points
which satisfy the equation.
That is, a point (x,y) is on the graph of an equation
if and only if x and y satisfy the equation.
Notice that the graph of an equation is a set of
points in the plane and therefore by definition, a
relation
Relations defined by Equations
Here, “x and y satisfy the equation” means “x and y
make the equation true”.
It is at this point that we gain some insight into the
word “relation”.
If the equation to be graphed contains both x and y,
then the equation itself is what is relating the two
variables.
Examples
More specifically, in the next examples, we will
consider the graph of the equation x2 + y3  1.
Even though it is not specifically spelled out, what
we are doing is graphing the relation
R  {(x,y) | x2 +y3  1}
Examples
The points (x,y) we graph belong to the relation R
and are necessarily related by the equation
x2 + y3  1, since it is those pairs of x and y which
make the equation true.
Determine whether or not (2,1) is on the graph of
x2 + y3  1.
Examples
To determine whether or not (2,1) is on the graph
of x2 + y3  1, we substitute x  2 and y  1 into
the equation to see if the equation is satisfied.
?
(2)  ( 1)  1
3 1
2
3
Hence, (2,1) is not on the graph of x2 + y3  1.
Examples
We now graph the equation x2 + y3  1.
To efficiently generate points on the graph of this
equation, we first solve for y in terms of x
x  y 1
2
3
y  1 x
3
3
2
3
y  1 x
2
y  1 x
2
3
3
Examples
We now substitute a value in for x, determine the
corresponding value y, and plot the resulting point
(x,y). For example, substituting x  3 into
the equation yields
y  1  x  1  (3)  8  2
3
2
3
2
3
so the point (3 , 2) is on the graph.
Examples
Continuing in this manner, we generate a table of
points which are on the graph of the equation. The
points are then plotted in the plane as shown below.
Examples
Remember, these points constitute only a small
sampling of the points on the graph of this
equation. To get a better idea of the shape of the
graph, we could plot more points until we feel
comfortable “connecting the dots”.
Doing so would result in a curve similar to the one
pictured below on the far left.
Examples
The graph x2 + y3  1 is given by
Intercepts of the graph
Definition: Suppose the graph of an equation is
given.
 A point on a graph which is also on the x-axis is called
an x-intercept of the graph.
 A point on a graph which is also on the y-axis is called
an y-intercept of the graph.
Intercepts of the graph
Finding the Intercepts of the Graph of an Equation
Given an equation involving x and y, we find the
intercepts of the graph as follows:
 x-intercepts have the form (x , 0); set y  0 in the
equation and solve for x.
 y-intercepts have the form (0 , y); set x  0 in the
equation and solve for y.
Domain and Range of a Relation
Suppose R is a relation. The sets of all x- and ycoordinates of the points in R are given special
names which we define below.
 The set of the x-coordinates of all the points in R is
called the domain of R.
 The set of the y-coordinates of all the points in R is
called the range of R.
Examples of Domain and Range
Find the domain and range of the following relations
and express them in interval notation whenever
possible
Functions
Introduction to Functions
One of the core concepts in College Algebra is that
of function. There are many ways to describe a
function and we begin by defining a function as a
special kind of relation.
Definition: A relation F in which each x-coordinate
is matched with only one y-coordinate is said to
describe y as a function of x.
Introduction to Functions
In other words, a function F is a relation with the
particular property that no two pairs of points in the
F have the same x-coordinate.
That is, in order to say y is a function of x, we just
need to ensure the same x-coordinate is not used in
more than one point.
Introduction to Functions
Which of the following relations describe y as a
function of x?
A quick scan of the points in R1 reveals that the x-coordinate 1
is matched with two different y-coordinates: namely 3 and 4.
Hence in R1, y is not a function of x, or, R1 is not a function.
Introduction to Functions
Which of the following relations describe y as a
function of x?
On the other hand, every x-coordinate in R2 occurs only once
which means each x-coordinate has only one corresponding
y-coordinate. So, R2 does represent y as a function of x, or, R2 is
a function.
Introduction to Functions
In order to see what the concept of function means
geometrically, we graph R1 and R2 in the plane.
Introduction to Functions
The fact that the x-coordinate 1 is matched with
two different y-coordinates
in R1 presents itself
graphically as the points
(1,3) and (1,4) lying on
the same vertical line, x  1.
Introduction to Functions
If we turn our attention to the graph of R2, we see
that no two points of the
relation lie on the same
vertical line.
Vertical Line Test
The Vertical Line Test: A set of points in the
plane represents y as a function of x if and only if
no two points lie on the same vertical line.
In other words, a relation R represents y as a
function of x if and only if no two points in R lie on
the same vertical line.
Vertical Line Test
Use the Vertical Line Test to determine which of
the following relations describes y as a function of
x.
Vertical Line Test
Use the Vertical Line Test to determine which of
the following relations describes y as a function of
x.
Domain and Range of a Function
Suppose F is a relation that defines y as a function
of x. That is, F is a function. Then, as before we
have,
 The set of the x-coordinates of all the points in F is
called the domain of F.
 The set of the y-coordinates of all the points in F is
called the range of F.
Domain and Range of a Function
Example: Find the domain and range of the
function
F {(3, 2), (0,1), (4,2), (5,2)}
Solution:
The domain of F is the set of the x- coordinates of
the points in F, namely, Dom F {3, 0, 4, 5}
and the range of F is the set of the y-coordinates,
Namely, Ran F {1, 2}
Domain and Range of a Function
Example: Find the domain and range of the
Function G whose graph is given below.
Domain and Range of a Function
To determine the domain and range of G, we need
to determine which x and y values occur as
coordinates of points on the given graph. To find
the domain, it may be helpful to imagine collapsing
the curve to the x-axis and determining the portion
of the x-axis that gets covered. This is called
projecting the curve onto the x-axis.
Domain and Range of a Function
Before we start projecting, we need to pay attention
to two subtle notations on the graph: the arrowhead
on the lower left corner of the graph indicates that
the graph continues to curve downwards to the left
forever more; and the open circle at (1,3) indicates
that the point (1,3) is not on the graph, but all
points on the curve leading up to that point are.
Domain and Range of a Function
Dom G {x | x <1}  (,1)
Domain and Range of a Function
Ran G {x | x 4}  (,4]
Remark about Functions
All functions are relations, but not all relations
are functions. Thus the equations which described
the relations in previous examples may or may not
describe y as a function of x.
The algebraic representation of functions is the
most important way to view them so we need a
process for determining whether or not an equation
of a relation represents a function.
Three Examples
Determine which equations represent y as a
function of x.
1. x3 + y2  1
2. x2 + y3  1
3. x2y  1  3y
Solution to Example 1
1. x3 + y2  1
Notice that in this case we are referring to the
relation defined by
R  {(x,y) | x3 +y2  1}
We solve for y and determine whether each choice
of x will determine only one corresponding value
of y.
Solution to Example 1
Solving for y in terms of x, we get
x  y 1
3
2
y  1 x
2
3
y 2  1  x3
y  1 x
3
y   1 x
3
Solution to Example 1
If we substitute x  0 into our equation for y, we
get y   1, so that (0,1) and (0,1) are on the
graph of this equation.
Hence, this equation does not represent y as a
function of x. Or equivalently, the relation
R  {(x,y) | x3 +y2  1}
is not a function.
Solution to Example 1
The graph of the relation
R  {(x,y) | x3 +y2  1}
is given by
Solution to Example 2
2. x2 + y3  1
Notice that in this case we are referring to the
relation defined by
f  {(x,y) | x2 + y3  1}
We solve for y and determine whether each choice
of x will determine only one corresponding value
of y.
Solution to Example 2
Solving for y in terms of x, we get
x  y 1
2
3
y  1 x
3
2
y  1 x
3
2
For every choice of x, the equation returns only one
value of y. Hence, this equation describes y as a
function of x.
Solution to Example 2
The graph of the function
f  {(x,y) | x2 + y3  1}
is given by
Solution to Example 3
3. x2y  1  3y
Notice that in this case we are referring to the
relation defined by
g  {(x,y) | x2y  1  3y}
We solve for y and determine whether each choice
of x will determine only one corresponding value
of y.
Solution to Example 3
x y  1 3y
2
Solving for y we get
x
2
 3 y  1
1
y 2
x 3
For every choice of x, the equation returns only one
value of y. Hence, this equation describes y as a
function of x.
Solution to Example 3
The graph of the function
g  {(x,y) | x2y  1  3y}
is given by
Function Notation
A function is a special kind of relation, one in
which each x-coordinate is matched with only one
y-coordinate. Here is another way of saying it:
A real-valued function f is a relation 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.
Function Notation
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
Function Notation
Any symbol that represents an arbitrary number in
the domain of a function f is called an independent
variable. Any symbol that represents a number in
the range of f is called a dependent variable.
A function, or relation in general, 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 or implied domain.
Algebraically Defined Function
Is a function represented by a formula. It has the
format y  f (x)  “expression in x”
2
f
(
x
)

3
x
 2 is a function.
Example:
f (5)  3(5)  2  77
Substitute 5 for x
f ( x  h)  3  x  h   2
Substitute x+h for x
2
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”
2
f
(
x
)

3
x
 2 is a function.
Example:
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.
s(t ) 
Example:
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
Notice that the table
x-values
x1
y = f (x) f (x1)
x2
…
…
xn
f (x2)
…
…
f (xn)
defining the function f is the relation
f {(x1 , f (x1)), (x2 , f (x2)), … (xn , f (xn))}
written in row format, where the first row is the
domain of f and the second row is the range of f.
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. Thus
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 human population of the world P
depends on the time 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
Implicit Form of a Function
Implicit Form
Explicit Form
F(x,y)  constant
y  f(x)
Graphs of Functions
Obtaining Information from or
About the Graph of a Function
Graphically Specified Function
In applications, the graph of a function often demonstrates
more clearly the relationship between the independent
variable x and the dependent variable y. Recall that,
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
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, (connecting the
successive points by line segments)
W(5) = 16
W(4.5) 
More Examples
Determine the domain, range, and intercepts of the function
defined by the following graph.
y
4
(2, 3)
(10,0)
0
(0, -3)
-4
(1, 0)
(4, 0)
x
More Examples
More Examples
Consider the function f (x)  x/(x+1).
1. Is the point (1,1/2) on the graph of f ?
2. If x  2, what is f (x) ? What point is on the graph
of f ?
3. If f (x)  2, what is x ? What point is on the graph
of f ?
Average Cost Function
The average cost function C of manufacturing x
computers per day is given by the function
1. Determine the average cost of manufacturing 30, 40,
and 50 computers per day.
2. Graph the function C(x) for 0 < x  80 and find the value
of x that minimizes the average cost
Finding Values of a Function
or
Evaluating a Function
Evaluating a Function
Example: Let f (x)  x2 + 3x + 4.
1. Find and simplify the following.
a)
b)
c)
d)
f (1), f (0), f (2)
f (2x), 2 f (x)
f (x + 2), f (x) + 2, f (x) + f (2)
(f (x + h) f (h))/h where h0
2. Solve f (x) = 4.
More Practice Problems
Is the Relation a Function?
Determine whether each relation represents a function.
If it is a function, state the domain and range.
1. {(2, 3), (4, 1), (3, -2), (2, -1)}
2. {(-2, 3), (4, 1), (3, -2), (2, -1)}
3. {(2, 3), (4, 3), (3, 3), (2, -1)}
Is the Relation a Function?
1
Determine if the equation y   x  3 defines y as a
2
function of x.
Determine if the equation x  2 y 2  1 defines y as a
function of x.
Evaluating a Function
Example: Let f (x)  3x2 + 2x.
1. Find and simplify the following.
a)
b)
c)
d)
f (1), f (0), f (1)
f (2x), 2 f (x),  f (x)
f (x + 1), f (x) + 1, f (x) + f (1)
(f (x + h)f (h))/h where h0
2. Solve f (x) = 5.
Find the Domain of the Function
Find the domain of each of the following functions.
Write the answer in interval notation
x4
f  x  2
x  2x  3
g  x  x  9
2
h  x   3  2x
The Algebra of Functions
or
Sum, Difference, Product,
and Quotient of Functions
Function Arithmetic
Suppose f and g are functions and x is in both the
domain of f and the domain of g.
 The sum of f and g, denoted f + g, is the function
defined by the formula
( f + g)(x)  f(x) + g(x)
 The difference of f and g, denoted fg, is the
function defined by the formula
( fg)(x)  f(x)g(x)
Function Arithmetic
 The sum of f and g, denoted fg, is the function
defined by the formula
( f g)(x)  f(x)g(x)
 The difference of f and g, denoted fg, is the
function defined by the formula
( fg)(x)  f(x)g(x)
provided g(x)  0.
Function Arithmetic
Notice that in the previous definitions, x is in both,
the domain of f and the domain of g. That is, the
domain of f + g, fg, fg, and fg is the
intersection of the domain of f and the domain of g.
In addition, in the case of the quotient fg, the
values of x that make g(x)  0 must also be
excluded.
Examples
For f(x)  2x2 +3 and g(x)  x3 +8 find f + g,
fg, fg, and fg and write their domains using
the interval notation.
( f + g)(x) 
( fg)(x) 
( f g)(x) 
( fg)(x) 
Examples
For f(x)  1/(x +2) and g(x)  x /(x 1) find f + g,
fg, fg, and fg and write their domains using the
interval notation.
( f + g)(x) 
( fg)(x) 
( f g)(x) 
( fg)(x) 