Download 1-1 Functions

1-1 Functions
Two ways to describe or define functions
Note the functions defined on p. 13:
Cost function: C = (fixed costs) + (variable costs)
C = a + bx
E. g.
C = 3000 + 200x
This is an example of a function defined by an equation, with
 independent variable x
 dependent variable C
When graphing such a function
 the independent variable labels the horizontal axis
 the dependent variable labels the vertical axis
Alternatively, the Cost function could have been given as
C(x) = 3000 + 200x
This is a function defined using functional notation. Here,
the “C” is the name of a function, not the name of a variable,
as in the prior example.
The independent variable is still x, but there is no dependent
variable, so you can label the vertical axis with “C(x)”.
Using this notations, we can write:
C(1000) = 3000 + 200(1000) = 203000
That is, the cost of producing 1000 units is $203,000.00
The 1000 is called an input, with corresponding output
203000.
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p.1
Some common business functions (see p. 13)
Cost:
C = a + bx
the cost of producing x items
The “parameters” (numbers that are specific to a particular
business situation), are a and b.
Example: C = 100 + .50x
fixed cost = $100
cost per item produced = $.50
Price-demand:
p = m - nx
x is the number of items that can be
sold for a price of $p per item
Here, m and n are the parameters that will vary according to
the business situation.
Example:
p = 1 - .0001x:
Price
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Demand
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Notice that as the price goes up, the demand will drop.
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p.2
Revenue: Revenue will be (#items sold) x (price per item)
R = xp
where
x = #items sold
p = price per item
Looks like as price increases, revenue increases, right?
But as price increases, demand (x) decreases, remember?
Two conflicting forces. How do they resolve?
For our example: R(x) = xp = x(1 - .0001x)
Here’s the Revenue graph for our example:
Revenue
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Demand
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Note: lowered prices  greater demand  increased sales
BUT the lower prices eventually overtake increased sales,
ultimately decreasing revenues.
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p.3
Profit:
Profit is, of course, Revenue – Cost: P = R – C
For our example:
C(x) = 100 + .50x
R(x) = x(1 - .0001x)
P(x) = x(1 - .0001x) – (100 + .50x) = -.0001x2 +.50x - 100
Here, Profit is written in terms of Demand (x).
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p.4
Guidelines for graphing functions
Label and scale your graphs!
(1) both axes must be labeled according to the names of the
variables given by the problem:
independent variable (often “x”) on the horizontal
axis
dependent variable (often “y”) on the vertical axis
if the graph is the graph of a named function, there is no
dependent variable; instead, the name of the function is
used; e.g. "f(x)", as shown above
(2) scaling should be done so as to convey the sense of scale
of the graph without overly cluttering it:
if the scale goes from 1 to 100, don't draw in 100 ticks and
scale each one
usually 2-10 scale ticks, with 2-3 of them labeled will
convey the scale, as shown above
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p.5