A relation is a set of ordered pairs where the first components of the ordered pairs are the input values and the second components are the output values.
A function is a relation that assigns to each input number EXACTLY ONE output number.
The domain is the set of all input values to which the rule applies. These are called your independent variables. These are the values that correspond to the first components of the ordered pairs it is associated with.
The range is the set of all output values. These are called your dependent variables. These are the values that correspond to the second components of the ordered pairs it is associated with.
Function Notation
f(x) read "f of x"
f is the function name. Output values are also called functional values. Note that you can use any letter to represent a function name, f is a very common one used.
x is your input variable.
Think of functional notation as a fancy assignment statement. When you need to evaluate the function for a given value of x, you simply replace x with that given value and simplify. For example, if we are looking for f(0), we would plug in 0 as the value of x in our function f.
If the function is constant, that means that the functional value never changes, it is always equal to that constant.
f(x) = c, where c is a constant.
Assignments
DIRECTIONS for Questions 1 and 2: Answer the questions based on the following information:-
A, S, M and D are functions of x and y, and they are defined as follows:
A(x, y) = x + y
S(x, y) = x - y
M(x, y) = xy
D(x, y) = x/y, where y in not equal 0.
1. What is the value of M(M(A(M(x, y), S(y,x)), x), A(y, x)) for x = 2, y = 3 ?
2. What is the value of S(M(D(A(a, b), 2), D(A(a, b),2)), M(D(S(a, b), 2), D(S(a, b),2))) ?
3. What is the value of 1.1!+2.2!+3.3!+--------+n.n! ?
4. Let g(x) be a function such that g(x + 1) + g(x − 1) = g(x) for every real x. Then for what value of p is the relation g(x + p) = g(x) necessarily true for every real x?
5. A function y = f(n) is defined, for all natural numbers, as the sum of the digits of n.
if k is a natural number such that f(f(f(f(k)))) = 1 and k > f(f(k)).f (f(f(k))) > 1 what is the least number of digits that k can have?
6. A function y = f(n) is defined, for all natural numbers, as the sum of the digits of n. for a natural number m, what is the value of f(f(m − f(m)))?
A set is a collection of things. Absolutely anything can be considered a set.
Below you'll see just a sampling of items that could be considered as sets:
- Your favorite clothes
- A coin collection
- The items in a store
- The English alphabet
- Even numbers
A set could have as many entries as you would like.
It could have one entry, 10 entries, 15 entries, or even an infinite number of entries.
On the next page you'll find out that a set could even have no entries at all!
For example, in the above list the English alphabet would have 26 entries, while the set of even numbers would have an infinite number of entries.
Each entry in a set is known as an element. We'll find out more about elements in the next section.
Sets are written using curly brackets ("{" and "}"), with their elements listed in between.
For example the English alphabet could be written as {a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z}
and even numbers could be {0,2,4,6,8,10,...} (Note: the dots at the end indicating that the set goes on infinitely)
A union of two or more sets is another set that contains everything contained in the previous sets.
Union is designated by the symbol ∪.
If A and B are sets then A ∪ B represents the union of A and B
Example:
A={1,2,3,4,5} B={5,7,9,11,13}
A ∪ B = {1,2,3,4,5,7,9,11,13}
The intersection of two (or more) sets is those elements that they have in common.
Intersection is designated by the symbol ∩.
So if A and B are sets then the intersection (the elements they both have in common) is denoted by A ∩ B.
Example:
A={1,3,5,7,9} B={2,3,4,5,6}
The elements they have in common are 3 and 5
A ∩ B = {3,5}
Subset
Let A be the set of objects that you own in your home
Let B be the set of objects that you own which are kept on the second floor of your home
Let C be the set of objects that you own which are kept in your bedroom [Note the bedroom is own the second floor]
Let D be the set of objects that you own which are kept in your bedroom nightstand
Now we could say D is contained within C, which in turn is contained within B, which in turn is contained within A.
This is the notion of a subset.
D is said to be a subset of C since it is completely contained within C (another way to think of this is every element of set D is also an element of set C).
C is said to be a subset of B since it is completely contained within B (another way to think of this is every element of set C is also an element of set A).
B is said to be a subset of A since it is completely contained within A (another way to think of this is every element of set D is also an element of set C).
The symbol for subset is Ì.
So D Ì C and C Ì B and B Ì A.
However if even one element of one set is not contained within the other then thy are not subsets.
If A were defined as {1,2,3,4,5} and B as {3,4,5,6} then B would not be a subset of A since
“6” Î B but 6 Ï A.
The symbol for “not a subset” is Ë.
We would write B Ë A.
