What is a function?
Representations of functions
Domain
Piecewise defined functions
Symmetry
§1.1–Four Ways to Represent a Function
Tom Lewis
Spring Semester
2014
What is a function?
Representations of functions
Domain
Outline
What is a function?
Representations of functions
Domain
Piecewise defined functions
Symmetry
Piecewise defined functions
Symmetry
What is a function?
Representations of functions
Domain
Piecewise defined functions
Symmetry
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
Symmetry
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
Symmetry
Example (Algebraic representation)
Let f (x ) = x 2 sin(x ). This function is represented algebraically.
What is a function?
Representations of functions
Domain
Piecewise defined functions
Symmetry
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
Symmetry
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
Symmetry
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
Symmetry
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)
Symmetry
What is a function?
Representations of functions
Domain
Piecewise defined functions
Symmetry
Definition (The absolute value function)
For each real number x , let |x | denote the distance from x to the
origin, 0.
Problem
Express |x | as a piecewise function.
What is a function?
Representations of functions
Domain
Problem
Consider problem 56 from page 21.
Piecewise defined functions
Symmetry
What is a function?
Representations of functions
Domain
Piecewise defined functions
Symmetry
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
Symmetry
Definition (Even and odd symmetry)
There are two elementary types of symmetry that a function may
exhibit.
Even A function f is said to be even if f (x ) = f (−x ) for
each x in the domain of f .
Odd A function f is said to be odd if f (−x ) = −f (x ) for
each x in the domain of f .
What is a function?
Representations of functions
Domain
Piecewise defined functions
Problem
Discuss the symmetry of the following functions:
• f (x ) = x 2
• f (x ) = x 3
• f (x ) = x + 4x 5 .
Problem
1. If f is an odd function and f (−3) = 5, then what is f (3)?
2. If f is an even function and f (4) = 16, then what is f (−4)?
Symmetry