Download Question 1: What is a function? A variable is a symbol that

Question 1: What is a function?
A variable is a symbol that represents a quantity that may vary. Typically a variable is a
letter of the alphabet. However, not all letters of the alphabet are necessarily variables.
For instance, in physics the letter c is a constant that represents the speed of light. It is
called a constant because the speed of light in a vacuum does not change. Letters in
other alphabets such as the Greek alphabet can also be variables or constants. The
letter  (pronounced pi) is a constant used in geometry that has a value that is
approximately 3.14157.
Mathematics allows us to describe relationships between quantities in a variety of ways.
In your mathematical experience, you have probably been exposed to formulas that
relate variables like x and y. There is nothing special about the letters x and y. We could
have used any letters in a formula. However, most algebra classes use the variables x
and y so we’ll start there and introduce other variables later that are appropriate in finite
mathematics.
A very simple formula that relates the variables x and y is
y  2x
Since this formula is solved for y, you might think of substituting a number in place of x
to obtain a value for y. For instance, a value of x = 3 corresponds to a value of y = 6,
y  2x
6  23
We think of this as inputting the value of x = 3 to obtain an output of y = 6. The variable
corresponding to the input, x, is called the independent variable. The variable
corresponding to the output, y, is the dependent variable. We use the term dependent to
emphasize the fact that the value for y depends on the input x. There is nothing special
about the letters used in the relationship or even which letter matches with the
independent or dependent variables. These terms are used to help describe the input
and output from a relationship between two variables. Consider the formula
2
x
y
2
Since this formula is solved for x, we would use it to take a value for y to compute a
value for x. In this case we would think of y as the independent variable and x as the
dependent variable. This means that an input of y = 6 corresponds to an output of x = 3.
These are the exact same values as y  2 x . The reason for this is that y  2 x can be
solved for x to give x 
y
:
2
y  2x
y 2x

2 2
Divide both sides by 2.
y
x
2
Whether we start with x and multiply by 2 to get y or start with y and divide by 2 to get x,
the relationship between x and y is the same. The only change is our perspective on the
independent and dependent variable. In each of these cases, the variable that is solved
for is the dependent variable.
The picture gets murkier if the equation is not solved for a variable. Suppose we have
the equation
y  2x  0
This equation is not solve for x or y. It is not clear whether x or y is the independent
variable. In a case like this, we can specify which variable is the independent variable.
The choice we make may reflect the quantity the variable represents or may simply be
our own personal choice.
If we decide to make the independent variable x, we need to solve the equation for y to
yield
y  2x 3
In this form, it is easy to input a value for x to calculate an output y. If we decide to make
the independent variable y, we need to solve the equation for x to yield
x
y
2
If we have a value for y, this equation can be used to calculate an output x. The variable
chosen for the independent variable is often up to the user. In other contexts the
independent variable is determined by the traditional choice made by practitioners in the
field.
Example 1
Write an Equation with a Specified Independent Variable
Suppose you have the equation 10 P  2Q  100 relating the quantity Q of
a product demanded by consumers and the price P in dollars. Write the
equation with P as the independent variable.
Solution To write 10 P  2Q  100 with a certain variable as the
independent variable, solve the equation for the other variable.
If P is to be the independent variable, solve the equation for Q:
10 P  2Q  100
2Q  10 P  100
Subtract 10P from both sides
2Q 10 P  100

2
2
Divide both sides by -2
Q  5P  50
Simplify
a. Write the equation with Q as the independent variable.
Solution If Q is to be the independent variable, solve the equation for P:
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10 P  2Q  100
10 P  2Q  100
Subtract 2Q to both sides
10 P 2Q  100

10
10
Divide both sides by 10
1
P   Q  10
5
Simplify
Economists traditionally choose the quantity Q to be the independent
variable when working with functions relating price and quantity. In this
format, we can see that as the quantity Q demand by consumers
increases, the price of the good must decrease to make it attractive to
the consumer.
Q
0
5
10
15
20
P
10
9
8
7
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The term function describes a special type of relationship between the independent and
dependent variable. These values for these variables are chosen from two sets called
the domain and range of the function. The values for the independent variable are
chosen from the domain and the values for the dependent variable are chosen from the
range.
A function is a correspondence between the independent and
dependent variable such that each value of the independent variable
corresponds to one value of the dependent variable.
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The phrase “a function of” is used to tell the user what the independent variable is. For
instance, the phrase “y as a function of x” indicates that the independent variable is x
and the dependent variable is y.
To determine if an equation describes a function, identify the independent and
dependent variable. Then determine if each value of the independent variable selected
from the domain of the correspondence matches with no more than one value of the
dependent variable. If there is no more than one match, then the correspondence is a
function.
Example 2
Determine If An Equation Represents a Function
Does the equation 10 P  Q  500 describe P as a function of Q?
Solution Since this example specifies P as a function of Q, we know
that the independent variable is Q and the dependent variable is P. To
make it easier to see how P and Q are linked, solve the equation for the
dependent variable P:
10 P  Q  500
10 P  Q  500 P   101 Q  50
Subtract Q from both sides
Divide both sides by 10
In this form, we can see that a value like Q  100 corresponds to one
value of P, P  40 . In fact, for any value of Q you get only one value P.
This means that this equation describes a function.
Example 3
Determine If An Equation Represents a Function
Does the equation x 2  y 2  4 describe y as a function of x?
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Solution This example specifies y as a function of x so we know that the
independent variable is x and the dependent variable is y. Like the
previous example, solve for the dependent variable y:
x2  y 2  4
y 2   x2  4
y    x2  4
Subtract x2 from both sides
Square root both sides
If we try a value like x  10 , the radicand becomes 102  4 or -96.
Assuming we are using real numbers, we can’t take the square root of a
negative number. The input to this equation is not a reasonable input
because it is not a part of the domain of this function.
To be a part of the domain of this function, the input to this function
needs to make the radicand nonnegative. The value of x must satisfy
 x 2  4  0 . Inspecting this inequality, we can see that  x 2  4  0 at x = -
2 or 2. Values between -2 and 2 like x = 0 make the expression  x 2  4
positive. This means the domain of this relationship is all real numbers
greater than or equal to -2, but less than or equal to 2.
If we pick values from the domain like x = 0, we get two outputs,
y  02  4  2 or y   02  4  2 . Since there is a number in the
domain that corresponds to more than one member of the range, the
relationship x 2  y 2  4 does not describe a function.
A graph of this equation verifies this conclusion. The graph of this
equation is a circle.
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Figure 1 – For this graph, any input between x = -2 and x = 2 leads to two outputs. This means the graph
does not correspond to a function.
On this circle, we can see that the line x = 0 crosses the graph at the
points (0, 2) and (0, -2). This means that the input x = 0 corresponds to
two different outputs. In fact, any input except for x = 2 or x = -2
corresponds to two outputs since vertical lines cross the graph in two
places.
In Example 2, we found that by writing 10 P  Q  500 in the form P   101 Q  50 we could
easily check to see if each value of Q led to no more than one value of P. This told us
that this relationship was a function since each value of the independent variable led to
no more than one value of the dependent variable. Now let’s look at this function a little
closer.
You may have thought that the variables P and Q were a bit strange. After all, in most
math textbooks you typically work with the variables x and y. In these situations you
were working with equations and their corresponding graphs in the x-y plane. You were
most concerned with the shapes of these graphs and the equations usually had little
basis in an application.
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In business and finance, every equation is based on an application. The names of the
variables often help you to understand what they represent. For instance, the variables
P and Q usually represent the price and quantity of some good. These variables can be
related to each other in one of two different ways. A demand function relates the price P
of a good to the quantity Q of the good demanded by consumers. The function
P   101 Q  50
is an example of a typical demand function. As Q gets larger, bigger and bigger
negative numbers are added to 50 resulting in smaller and smaller values. A graph of
this function reflects this characteristic. In Figure 2, the graph is a line that drops as you
move from left to right. This means that as the quantity is increased (move left to right),
the price drops.
Figure 2 – As the quantity Q increases from 0 to 500 horizontally, the price P
drops from 50 to 0. This means that the quantity demanded by consumers
increases as the price drops.
This function’s graph is a straight line. A function whose graph is a straight line is called
a linear function.
Looking at the graph, it is easy to recognize a linear function. We would also like to be
able to recognize a linear function from its equation.
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Any equation that can be written in the form
y  mx  b
is a linear function. In this form, we say that y is a linear function of x.
The letters m and b are real numbers corresponding to constants and
x and y are variables.
The graph of this equation is a straight line. The slope of the line is
m. The y-intercept of the line is b
It is easy to read this definition without examining how it applies to the line
P   101 Q  50 . The equations y  mx  b and P   101 Q  50 may look different, but are
really very similar.
The definition for a linear function contains four letters: y, m, x, and b. Some of these
letters are variables and others are constants. To insure that the user of this function
knows which letters are variables and which letters are constants, we need to define the
variables.
One way of doing this is to write, y is a linear function of x. By writing this phrase, we
know that the variables in the equation y  mx  b are x and y. All other letters are
constants representing numbers.
By modifying the phrase, we can write other linear functions with different variables. The
phrase “P is a linear function of Q” corresponds to the equation
P  mQ  b where Q is the independent variable, P is the dependent variable and m and b are
constants. An example of a linear function in which P is written as a linear function of Q
is
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P   101 Q  50 In this case, the value of m is  101 and b is 50.
Example 4
Does An Equation Correspond to a Linear Function?
Decide if the equation 5 P  10Q  100 can be written so that P is a linear
function of Q.
Solution To decide if the equation can be written so that P is a linear
function of Q, we need to rewrite the equation in the form P  mQ  b .
By stating “linear function of Q”, the example implies that the variable on
the right side is Q. Solve the equation for P to put the equation in this
form:
5P  10Q  100
5P  10Q  100 P  2Q  20
Subtract 10Q from both sides
Divide each term by 5
The equation can be written in the proper form with m  2 and b  20 .
Example 5
Does An Equation Correspond to a Linear Function?
Decide if the equation 5 P  10Q  100 can be written so that Q is a linear
function of P.
Solution Compared to Example 4, this example reverses the role of the
variables. “Q is a linear function of P” means that we want to rewrite the
equation in the form Q  mP  b . To accomplish this task, we must solve
the equation for Q:
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5P  10Q  100
10Q  5P  100 Q   12 P  10
Subtract 5P from both sides
Divide each term by 10
The equation can be written in the proper form with m   12 and b  10 .
Example 4 and Example 5 suggest that an equation must be solved for the dependent
variable to determine if the equation is a linear function. The phrase
Dependent Variable is a linear function of the Independent Variable
allows you to obtain the appropriate form for the linear function. The dependent variable
is always written first in this statement and the independent variable is always written
after “linear function of”. Based on this phrase, we know the form will be
Dependent Variable  m  Independent Variable  b An equation that can be written in this form, by solving for the dependent variable, is a
linear function. If a function cannot be written in this form, it is not a linear function.
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