Transcript 1.1 Functions

1.1 Functions
• This section deals with the topic of
functions, one of the most important topics
in all of mathematics. Let’s discuss the
idea of the Cartesian coordinate system
first.
Cartesian Coordinate System
• The Cartesian coordinate system was named after
Rene Descartes. It consists of two real number
lines which meet at a point called the origin. The
two number lines which meet at a right angle
divide the plane into four areas called quadrants.
• The quadrants are numbered using Roman
numerals as shown. Each point in the plane
corresponds to one and only one ordered pair of
numbers (x , y). Two ordered pairs are shown.
I
II
III
(3,1)
x
(-1,-1)
y
IV
Graphing an equation
• To graph an equation in x and y, we need to find ordered
pairs that solve the equation and plot the ordered pairs on
a grid.
For example, let’s plot the graph of the equation
y = x2 + 2
Making a table of ordered pairs
• Let’s make a table of ordered pairs that
satisfy the equation y = x2 + 2
x
-3
-2
-1
0
1
2
y
Plotting the points
• Next, plot the points and connect them with a smooth
curve. You may need to plot additional points to see the
pattern formed.
Function
• The previous graph is the graph of a function. The idea of
a function is this: a relationship between two sets D and R
• such that for each element of the first set, D, there
corresponds one and only one element of the second set, R.
For example, the cost of a pizza (C) is related to the size of
the pizza. A 10 inch diameter pizza costs 9.00 while a 16
inch diameter pizza costs 12.00.
Function definition
• You can visualize a function by the following diagram which shows a
correspondence between two sets, D, the domain of the function and
R, the range of the function. The domain gives the diameter of pizzas and
the range gives the cost of the pizza.
range
domain
10
12
16
9.00
10.00
12.00
Functions specified by equations
• Consider the previous equation that was
graphed
-2
Input x = -2
Process: square (–2)
then subtract 2
(-2,2) is an ordered
pair of the function.
2
Output: result is 2
Function Notation
• The following notation is used to describe functions The
variable y will now be called
• This is read as “ f of x” and simply means the y
coordinate of the function corresponding to a given x
value.
Our previous equation
can now be expressed as
Function evaluation
• Consider our function
• What does
mean?
Replace x with the
value –3 and evaluate the expression
• The result is 11 . This means that the point (-3,11) is on
the graph of the function.
Some Examples
• 1.
f (a)  3(a)  2
f (6  h)  3(6  h)  2  18  3h  2
 16  3h
Domain of a Function
• Consider
f ( x)  3x  2
f (0)  ?
f (0)  3(0)  2  2
• which is not a real number. Question: for
what values of x is the function defined?
Domain of a function
• Answer:
• is defined only when the radicand (3x-2)
• is greater than or equal to zero. This implies that
3x-2
0
or
2

x 3
Domain of a function
• Therefore, the domain of our function is the set of real
numbers that are greater than or equal to 2
3
• Examples. Find the domain of the following functions.
1
f ( x) 
x4
2
• Answer:
 x x  8 , [8, )
More examples
• Find the domain of
1
f ( x) 
3x  5
• In this case, the function is defined for all values of x
except where the denominator of the fraction is zero.
This means all real numbers x except 5
3
Mathematical modeling
• The price-demand function for a company is given by
•
p( x)  1000  5 x,
•
0  x  100
where P(x) represents the price of the item and x
represents the number of items. Determine the revenue
function and find the revenue generated if 50 items are
sold.
Solution
• Revenue = price x quantity so
• R(x)= p(x)*x = (1000  5 x) x
• When 50 items are sold, x = 50 so we will evaluate the
revenue function at x = 50
R(50)  (1000  5(50)) 50  37,500
•
The domain of the function has already been specified.
We are told that
0  x  100