What is a function?
Representations of functions
Domain
Piecewise defined functions
Difference quotients
§1.1–Four Ways to Represent a Function
Tom Lewis
Fall Semester
2015
What is a function?
Representations of functions
Domain
Outline
What is a function?
Representations of functions
Domain
Piecewise defined functions
Difference quotients
Piecewise defined functions
Difference quotients
What is a function?
Representations of functions
Domain
Piecewise defined functions
Difference quotients
Definition (Function)
• A function is a rule that assigns to each element x in a set A
exactly one element, called f (x ), in a set B .
• The variable x is called the independent variable. The variable
y is called the dependent variable.
• The set A is called the domain of the function f ; it is the set
of admissible inputs of the function.
• The set B is called the range of the function f ; it is the set of
outputs of the function.
What is a function?
Representations of functions
Domain
Piecewise defined functions
Difference quotients
Problem
On the first day of kindergarten the children are accompanied by
their (biological) mothers.
1. Is matching mothers to their children a functional relationship?
2. Is matching children to their mothers a functional relationship?
What is a function?
Representations of functions
Domain
Piecewise defined functions
Difference quotients
Example (Algebraic representation)
√
Let f (x ) = x 2 1 + x 4 . This function is represented algebraically.
What is a function?
Representations of functions
Domain
Piecewise defined functions
Difference quotients
Example (Visual representation)
A water tap is opened and the water is allowed to flow freely for 60
minutes. The graph of the temperature as a function of time is
given below.
Temperature (F)
150
125
100
75
50
25
10
20
30
40
Time (minutes)
50
60
What is a function?
Representations of functions
Domain
Piecewise defined functions
Difference quotients
Example (Verbal representation)
Let P (t ) represent the number of people alive on the planet at
time t (measured in seconds since the year 1 AD). This function is
represented verbally.
What is a function?
Representations of functions
Domain
Piecewise defined functions
Difference quotients
Example (Numerical representation)
In the table, the variable y is a function of x :
x
0
1
2
3
y
.25
.36
.78
1.1
The functional relationship is represented numerically, by the table
of values.
What is a function?
Representations of functions
Domain
Piecewise defined functions
Difference quotients
Problem
Find the (natural) domain of the function in each case:
√
1. f (x ) = 5 − x
5+t √
2. h(t ) =
+ t
8−t
√
s + 10
3. g(s) = √
8−s
What is a function?
Representations of functions
Domain
Piecewise defined functions
Example
Define a function piecewise as
f (x ) =
−x 2
if x 6 0
x
if x > 0
Evaluate (if possible) the following:
1. f (−1)
2. f (2)
3. f (0)
Difference quotients
What is a function?
Representations of functions
Domain
Piecewise defined functions
Difference quotients
Piecewise defined functions
Difference quotients
Problem
Consider problem 56 from page 22.
What is a function?
Representations of functions
Domain
Problem
Let f (x ) = x 2 + 3x and let h 6= 0. Simplify the expression
f (4 + h) − f (4)
.
h
Note: this is an important problem!
What is a function?
Representations of functions
Domain
Piecewise defined functions
Problem
Let f (x ) = 1/x and let a 6= b. Simplify the expression
f (a) − f (b)
.
a −b
Difference quotients