Download Examples of Functions - Stony Brook Mathematics

Justin Rathjen
MAE301
Notes On 9/17/07
9/18/07
Definition: A Function is a mapping
F: A  B such that every element a in A, maps to exactly one element of B.
The variable x is where a number comes into the function. Therefore, we could call x the
input variable. The function rule says to multiply the number in x by 3 and then put this
result, or output, into the variable y. The variable y, therefore, could be called the output
variable. The group, or set, of numbers that "goes into" a function is called the DOMAIN
of the function. The set of numbers that "comes out of" a function is called the RANGE
of the function. In this example, since x could accept any real number, we would say that
the domain of this function is all real numbers.
Equations of Functions:
1st Case – Where each output comes from only input.
Y = 3x………… Eg. (1, 3), (2, 6)
2nd Case - Two or more different inputs(x) yields the same output(y).
Y= x^4…………Eg. (2, 16), (-2, 16)
3rd Case - The equation is NOT a function if one unique input(x) can get you to two or
more different outputs(y).
Y^2 = x…………Eg. (4, 2), (4,-2)
Examples of Functions
1.
2.
3.
4.
5.
6.
7.
8.
Multiply by two. Y = 2x
Squaring. F (x) = x2.
Input: time walked, Output: distanced walked
Input: time since it began raining, Output: amount of rain which has fallen
Input: F, Output: C.  Inverse: Input: C, Output: F
Exponential growth. V=2000(1.04)t. Input: t = Time in years the money is on
deposit. Output: V = Value of the investment (dollars)
Profits. Input: time, # of months since company’s founding. Output: monthly
profits (dollars).
Input: time. Output: temperature.
Justin Rathjen
MAE301
Notes On 9/17/07
9/18/07
Injective Functions

A function f is said to be injective if, for every y in the co-domain, there is at most
one x in the domain such that f(x) = y.

f is injective if f (a) = f (b) implies a = b (or a
b in the domain. (1)
Injective
b implies f (a)
f(b)), for any a,
A non-injective function.
Surjective functions
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

A function f is said to be surjective if its values span its whole co-domain; that is,
for every y in the co-domain, there is at least one x in the domain such that
f(x) = y.
A function f: X→Y is surjective if and only if its range f (X) is equal to its codomain Y. A surjective function is called a surjection, and said to be onto. (1)
Justin Rathjen
A surjective function.
MAE301
Notes On 9/17/07
9/18/07
A non-surjective function.
Bijective functions



A function f from a set X to a set Y is said to be bijective if and only if for every y
in Y there is exactly one x in X such that f(x) = y.
Said another way, f is bijective if and only if it is a one-to-one correspondence
between those sets; i.e., both one-to-one (injective) and onto (surjective). (1)
A bijective function
Justin Rathjen
MAE301
Notes On 9/17/07
9/18/07
In geometry, a convex polygon is a simple polygon whose interior is a convex set.
The following properties of a simple polygon are all equivalent to convexity:







Every internal angle is at most 180 degrees.
Every line segment between two vertices of the polygon does not go exterior to
the polygon (i.e., it remains inside or on the boundary of the polygon).
A simple polygon is strictly convex if every internal angle is strictly less than 180
degrees.
If a simple polygon is not convex, it is called concave. At least one internal angle
of a concave polygon is larger than 180 degrees. (2)
Definition of Group Operation: Is a function from G x G  G, that takes (g1, g2) in G x
G, to g1 * g2 in G.
A group (G, *) is a set G with a binary operation * that satisfies the following four
axioms:
Closure : For all a, b in G, the result of a * b is also in G.
Associativity: For all a, b and c in G, (a * b) * c = a * (b * c).
Identity element: There exists an element e in G such that for all a in G, e * a = a * e = a.
Inverse element: For each a in G, there exists an element b in G such that a * b = b * a =
e, where e is an identity element.
Justin Rathjen
MAE301
Notes On 9/17/07
9/18/07
A sequence is a function whose domain is the natural numbers. Instead of using the f(x)
notation, however, a sequence is listed using the an notation. There are infinite sequences
whose domain is the set of all positive integers, and there are finite sequences whose
domain is the set of the first n positive integers.
Defining a Sequence
There are two common ways to define a sequence by specifying the general term.
- The first is to use a form that only depends on the number of the term, n. To find
the first five terms when you know the general term, simply substitute the values 1, 2, 3,
4, and 5 into the general form for n and simplify. (4)
General Term an
an = 3n-2
an = 1 / n
First Five Terms
3(1)-2=1; 3(2)-2=4; 3(3)-2=7; 3(4)-2=10; 3(5)-2=13
1, 4, 7, 10, 13
1/1, 1/2, 1/3, 1/4, 1/5
- The second way is to recursively define a sequence. A recursive definition uses
the current and/or previous terms to define the next term. You can think of ak+1 being the
next term, ak being the current term, and ak-1 being the previous term. (4)
Recursive Definition
a1 = 5, ak+1 = 2 ak - 1
First Five Terms
5 (by definition),
2(5) - 1 = 9 (twice the first term of 5 minus 1),
read that last part as "the next 2(9) - 1 = 17 (twice the second term of 9 minus 1),
2(17) - 1 = 33 (twice the third term of 17 minus 1),
term is one less than twice
2(33) - 1 = 62 (twice the fourth term of 33 minus 1)
the current term"
a1 = 2, a2 = 1,
2 (by definition),
ak+2 = 3ak - ak+1
1 (by definition),
3(2) - 1 = 5 (3 times first term minus second term),
read that last part as "the next 3(1) - 5 = -2 (3 times second term minus third term),
term is 3 times the last term 3(5) - (-2) = 17 (3 times third term minus fourth term)
minus the current term"
A group homomorphism is a map
between two groups such that the group
operation is preserved:
for all
, where the product on the
left-hand side is in and on the right-hand side in .
Note that a homomorphism must preserve the inverse, because
, so
.
Justin Rathjen
MAE301
Notes On 9/17/07
9/18/07
References:
www.groovyweb.uklinux.net/index.php?page_name=Injective%20Surjective%20and%20
Bijective%20functions&category=maths (1)
www.cs.cmu.edu/~eugene/research/full/planar-visibility.pdf (2)
www.mathworld.wolfram.com/GroupHomomorphism.html (3)
www.richland.edu/james/lecture/m116/sequences/sequences.html (4)