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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
Some common business functions (see p. 13)
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
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.
p = 1 - .0001x:
Notice that as the price goes up, the demand will drop.
Revenue: Revenue will be (#items sold) x (price per item)
R = xp
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:
Note: lowered prices  greater demand  increased sales
BUT the lower prices eventually overtake increased sales,
ultimately decreasing revenues.
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).
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
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