Transcript Section 1.1 - Models and Functions

MAT 213
Brief Calculus
Section 1.1
Models, Functions and Graphs
Mathematical Models
• The process of translating a real world problem
situation into a usable mathematical equation is
called mathematical modeling
• For example, in business it is important to know
how many units to produce to maximize profit
– Thus if we can model our profits as a function of the
number of units produced, we can use calculus to
determine how many products will maximize our profit
A function is a rule that assigns exactly one
output to every input
• In mathematics, a function is often used to
represent the dependence of one quantity
upon another
• We therefore define the input as the
independent variable, and the resulting
output as the dependent variable
• Note: the output does not have to depend on
the input in order to have a function
Definitions
Independent Variable
 Its values are the elements of the DOMAIN
 Plotted on the horizontal axis
 Its values are known when collecting data
Dependent Variable
 Its values are the elements of the RANGE
 Plotted on the vertical axis
 The quantity measured for a specific value of the independent
variable.
Is it a function???
Input
-2
-1
0
1
2
Output
9
7
5
3
1
Is it a function???
Input
5
5
5
5
5
Output
-2
-1
0
1
2
Is it a function???
Input
-2
-1
0
1
2
Output
5
5
5
5
5
What about these?
A = {(0,4), (7,4), (5,3), (1,0)}
B = {(0,1), (1,1), (1,0)}
C = {(1,1)}
Is it a function???
10
5
-10
-5
5
-5
-10
10
Is it a function???
10
5
-10
-5
5
-5
-10
10
Function Notation
We use the notation f(x) to denote a function.
It is read "f of x," meaning the value of the function f evaluated at
point x
Actually, any combination of letters can be used in function notation
Example: If we were writing a function that
described the area of a square in terms of
the length of a side, we may choose A(s)
to mean the area A when the side is length s.
The parentheses DO NOT mean multiplication!!!
Examples
Find the function values.
Do not worry about simplifying right now
h(x) = x2 + 2x - 4
a. h(4)
b. h(-3x)
c. h(a – 1)
d. h(x+1) – 3h(x)
g(x)
1. g(-2) = ?
2. g(-1) = ?
3. Find the values of x
that make g(x) = 0.
Rule of Four
•
Functions can be represented in 4 ways
1.
2.
3.
4.
Numerical data such as a table
Graphically
In words
By an equation
We will encounter all 4 of these
representations during the semester
In Business
Fixed costs (overhead)
Variable costs
Total Cost = Fixed costs + Variable costs
Total Cost
Average cost = ----------------------------Number of Units Produced
Profit = Revenue – Total Cost
When does a company break even?
Break-Even Point
$
Revenue
Total Cost
10
20
30
Number of Units
(in millions)
How many units would
this business need to
sell in order to break
even?
Break-Even Point
$
Profit
5
10
15
# of units
(in thousands)
How many units would
this business need to
sell in order to break
even?
Combinations of Functions
• Now if we have a revenue function, R(x), and a cost
function, C(x), we saw that we can create a profit
function, P(x)
• We would get P(x) = R(x) - C(x)
• Thus we have combined two functions via subtraction to
get another function
• We can also add, multiply or divide two functions
Composition of Functions
Notation
Take the functions f(x) and g(x)
f(g(x)) = (f◦g)(x)
To evaluate f(g(x)), always work from the inside out. First
find g(x) then plug that result into f.
For all x in the domain of f such that
f(x) is in the domain of g
Composition of Functions
Example
Let f(x) = 5x + 1
and g(x) = x2
Evaluate the following:
(f◦g)(x)
(f◦f)(x)
f(g(-2))
g(f(x))
In groups let’s try the following from
the book
• 1, 13, 25, 27, 35, 53