1.3.1 Functions and Their Representations I
Function: a mathematical idea
Informal definition of function
A function is a
rule of association
that associates any valid input
with exactly one output
the set of all possible valid inputs is called:
the domain of the function
the set of all possible outputs is called:
the range of the function
1.3.1-1
Function names and functional notation
In the following, 20 = input, 400 = output
Which is easier to write?
(1) "If the length of one side of a square room is 20 feet,
the area is 400 square feet"
(2) area(20) = 400
Functional notation
name of
input
function
area ( 20 )
output
input = 20  output = 400 : thus, area(20) = 400
input = 10  output = 100 : thus, area(10) = 100
whenever you see "area(10)" you can substitute 100
Don't confuse:
area(20) with
area x 20
 here, parens play a special functional notation role,
 not their more familiar grouping role, as in a(b + c)
 you will distinguish between the two roles by context
1.3.1-2
Representation 1: Defining a function
symbolically (using a formula)
Function name:
Definition:
Read:
Rule of association:
square
square(x) = x2
"square of x equals x squared"
the "x" on the left-hand side stands for the input
the "x2" is the formula used to compute the output
so square associates any input "x" with the output "x2"
whatever x is
Examples of usage:
square(2) =
square( 2 ) =
square(y) =
F(C) = (9/5)C+32
F(30) =
F(a) =
It's a no-brainer!
for any function defined by a formula,
computing the output from the input
is a matter of BRUTE FORCE SUBSTITUTION
f(x) = x2 + 2x + 3
f(3) = (3)2 +2(3) + 3 = 18
f(x + 2) = (x + 2)2 + 2(x + 2) + 3 =
= x2 + 4x + 4 + 2x + 4 + 3 = x2 + 6x + 11
f(x) - f(x + h) = (x2 + 2x + 3) - ((x + h)2 + 2(x + h) + 3)
= (x2 + 2x + 3) - (x2 + 2xh + h2 + 2x + 2h + 3)
= -2xh - h2 - 2h
1.3.1-3
Finding the implied domain of a symbolically
defined function
The definition for such a function may state explicitly what
its domain is. Otherwise, its implied domain is:
the set of all real numbers
for which the formula
evaluates to a real number
f(x) = x2
Domain f = ??
2
Domain f = ??
x 3
Note: the answer to f(3) = ? is “undefined”
f(x) =
f(x) =
x 5
Domain f = ??
Why? Square root of a number is real only if
the number is  0. So we must have:
x-50
or
x5
Domain f: { x | x  5 }
This is an example of set-builder notation. Read:
“the set of all x such that x is greater than or equal to 5”
Abbreviated version: just “x  5” will be an acceptable
answer.
1.3.1-4