Download What is function notation?

Question 2: What is function notation?
Problems in business and finance are often mathematical in nature. These problems
come from real-world situations that can be extremely complex. A mathematical model
is a mathematical representation of the situation. Often these representations take the
form of functions.
The price of a product may be constant or it may change in response to the quantity
sold. If the quantity of some product sold increases, the price may decreases. A
demand function displays the relationship between the price per unit P of a product and
the quantity Q demanded by consumers. For example, the relationship for a certain
product may be
P  0.0002Q  55 We say that the price P is a function of the quantity Q. By saying “function of quantity
Q”, we are indicating that the independent variable is Q and the dependent variable is P.
The same relationship can also be written as
Q  5000 P  275000 Now the quantity Q is a function of the price P. This means that P is the independent
variable and Q is the dependent variable.
Function notation is used to emphasize the independent variable in a function.
Functions are named with a letter or phrase. Next to the name is a set of parentheses
with the independent variable inside. For a demand function, we might use the letter D
to name the function. For the two equations above, we would write
D  Q   0.0002Q  55 or D  P   5000 P  275000 Knowing what the independent variable is helps us to determine the input to and out
from the function. For instance, if we need to find the price of the product when the
quantity demanded is 10,000 units we would set Q  10000 in D  Q   0.0002Q  55 . We
indicate this by writing
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D 10000   0.0002 10000   55  53 In the function notation, each Q is set equal to 10,000. Since the input is a quantity, the
output must be the corresponding price.
It may seem tedious to replace P with D  Q  to define a function. However, in many
situations we need to work with two equations simultaneously. Without a name for the
function, we would have a difficult time distinguishing the two formulas.
Example 6
Supply Functions
A supply function also displays the relationship between the price and
the quantity of a product. Unlike the demand function, the supply
function models the price P at which suppliers are willing to supply Q
units of the product. Suppose this relationship is
P  0.0001Q
a. Use function notation to define the supply function as a function of
the quantity Q. Use the letter S to name the function.
Solution Using the name S and the indepenent variable Q, the
appropriate function notation is S  Q  . The function definition is
S  Q   0.0001Q
b.
Use the function definition to find the price P at which suppliers
would be willing to supply 20,000 units.
Solution Set Q  20000 in the function to yield
S  20000   0.0001 20000   2 14
At a price of $2 per units, suppliers would be willing to supply 20,000
units.
c.
What does it mean for S  Q   20 ?
Solution When the function notation is set equal to a number, the output
from the function is being given. In this case, the price is equal to $20. If
replace the function notation with its formula, we can solve the resulting
equation to find the quantity Q corresponding to a price of $20:
0.0001Q  20
Q
20
0.0001
 200000
At a price of $20 per unit, suppliers are willing to supply 200,000 units.
Typically, the demand and supply functions for a product are graphed together. Let’s put
the two functions we have been using together.
D (Q )  0.0002Q  55
S (Q )  0.0001Q
Figure 3 – The equilibrium point is the point of intersection of the demand and supply functions.
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On this graph, the two functions cross. At the point of intersection, the quantity that
consumers are willing to buy is equal to the quantity that manufacturers are willing to
supply. We can find this point algebraically by setting the two functions equal,
D  Q   S  Q  and solving the resulting equation for Q.
Example 7
Find the Equilibrium Point
The supply and demand functions for robotic hamsters are
D  Q   0.0002Q  55
S  Q   0.0001Q
Find and interpret the equilibrium point.
Solution The supply and demand are equal when
0.0001Q  0.0002Q  55
0.0003Q  55
Q
55
 183333
0.0003
The price at this quantity may be obtained from either function. The
equilibrium price is
S 183333  0.0001183333  18.33 dollars
At this price, the quantity demanded by consumers and the quantity
manufacturers are willig to supply is approximately 183,333 robotic
hamsters.
Businesses operate by obtaining money by the sales of goods and services. The
amount of money obtained through the sales of goods and services is called revenue.
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Revenue is modeled by multiplying the quantity a good or service by the price of each
unit of the good or service. We can write this model in mathematical form by writing
revenue  price per unit  quantity In some situations, the price per quantity may be a constant. For the robotic hamsters,
the price per quantity is given by the demand function D (Q ) . When we substitute the
demand function into the revenue relationship, we get the revenue function,
R  Q   D  Q  Q We can form this function using the demand function.
Example 8
Revenue Function
The demand function for robotic hamsters is
D  Q   0.0002Q  55
a. Use function notation to define the revenue function for this
application as a function of the quantity Q.
Solution We need to write a revenue function, R as a function of Q,
R(Q) . Using the revenue relationship, we find
R Q   D Q  Q
  0.0002Q  55  Q
 0.0002Q 2  55Q
b. Use the function from part a to determine the revenue when 15,000
robotic hamsters are produced and sold.
Solution The revenue at Q  15000 is
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R 15000   0.0002 15000   55 15000 
2
 780000
Since the demand function returns a price in dollars per unit and the
quantity is in units, the product is in dollars. The revenue from 15,000
robotic hamsters is $780,000.
The revenue function in Example 8 is an example of a quadratic function. Quadratic
functions include a term in which the variable is squared. This makes them different
from a linear function that contains constants and terms where the variable is raised to
the first power.
Any equation that can be written in the form
y  ax2  bx  c
is a quadratic function. In this form, we say that y is a quadratic
function of x. The letters a, b, and c are real numbers corresponding
to constants and x and y are variables. In addition, a must not equal
zero.
The graph of a quadratic function is a parabola. If a  0 , the parabola
has a low point on it. If a  0 , the parabola has a high point.
We can use quadratic functions to model other economic functions like profit.
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The name of a function is arbitrary, but care needs to be taken so that names are not
confusing. Quantities beginning with the letter p are especially problematic. Two
different economic quantities begin with the letter p, price and profit. To distinguish
between them, we’ll need to name them carefully.
Profit is the difference between revenue and cost. We can write this mathematically as
profit  revenue  cost
If the amount received from sales is greater than the cost, the profit is positive since the
revenue is greater than the cost. On the other hand, if the costs are greater than the
revenue, the profit is negative.
To name a profit function with an independent variable Q, we might want to write P(Q) .
Although this is perfectly acceptable, the name P might be confused with the variable P
representing price. To avoid this confusion, it would be wise to use the name Profit(Q) .
This function takes the quantity Q of some good or service and outputs the profit at that
production level.
If a word is used to name a function instead of simply a letter, we should probably
continue this pattern with other related function. Instead of R(Q) for the revenue
function, we could use the name Revenue(Q) . Instead of C (Q) for the cost function, we
could use the name Cost(Q) . The names are very descriptive of exactly what the
function does and allow us to write the relationship between these functions as
Profit(Q)  Revenue(Q)  Cost(Q) The name of a function is up to the user. Some textbooks might choose to use an entire
world while others might use a single letter. We’ll use both naming conventions so you
get used to them.
Example 9
Profit Function
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The cost of producing robotic hamsters at an Asian manufacturing plant
is
Cost(Q)  5Q  750000 dollars
where Q is the number of robotic hamsters. The revenue from selling
the toys is
Revenue(Q)  0.0002Q2  55Q dollars .
a. Find the profit function.
Solution The profit function is formed by subtracting Cost(Q) from
Revenue(Q) ,
Profit(Q)   0.0002Q 2  55Q    5Q  750000 
 0.0002Q 2  55Q  5Q  750000
 0.0002Q 2  50Q  750000
This function is a quadratic function of Q with a  0.0002 , b  50 , and
c  750000 .
b. Find the profit at a production level of 100,000 robotic hamsters.
Solution Substitute Q equal to 100,000 into the profit function,
Profit(Q)  0.0002Q2  50Q  750,000 , to yield
Profit(100000)  0.0002 100000   50 100000   750, 000
2
 2250000
At a production level of 100,000 robotic hamsters, the profit is
$2,250,000.
c.
Does the profit function have a high point or low point?
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Solution For this parabola, a  0 . This mean the graph has a high point.
In later chapters we’ll learn how to find the high and low points on any function using
calculus. Finding these points is one of the useful applications for calculus.
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