•In this lesson I've learned that the FUNCTIONS are relation between a set of inputs and a set of outputs with the property that each input maps to exactly one output.

•RELATION is a pairing of input values with output values. It can be shown as a set of ordered pairs (x,y) where x is an input and y is an output.            

(ILLUSTRATIVE)          

"DOMAIN"

•The DOMAIN of a function is the complete set of possible values of the independent variable.

•The DOMAIN is the set of all possible x-values which will make the function "work", and will output real y-values.

The RANGE of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain.

ORDERED PAIR = (x,y)

  Where, x = abscissa, the distance measure of a point from the primary axis “x”

  And, y = ordinate, the distance measure of a point from the secondary axis “y”

•A ONE-TO-ONE function is a function in which the answers never repeat. A normal function can have two different input values that produce the same answer, but a one-to-one function does not.

•Having the property that the same argument may yield multiple values, but different arguments never yield the same value 

•A MANY-TO-ONE relation associates two or more values of the independent (input) variable with a single value of the dependent (output) variable. 

A function is a relation in which each input has only one output.

In the relation , y is a function of x, because for each input x (1, 2, 3, or 0), there is only one output y. x is not a function of y, because the input y = 3 has multiple outputs: x = 1 and x = 2.

In this chapter,I learn how to perform basic operations with functions--addition, subtraction, multiplication, and composition--as well as how to find the inverse of a function.

The first method involves substituting x for f (x), substituting f-1(x) for x, and isolating f-1(x). The second method involves graphing the function and reflecting it over the line y = x. All three methods of finding inverses are useful, and they can be used in combination to check each other.

In this composition of functions I have learned about a function is like a machine. It performs a set of operations on an input in order to produce an output. Therefore, a composition of functions occurs when the output, or result, of one function becomes the input of another function.

For functions represented by f(x) or g(x), the composition would be represented by f(g(x)) or g(f(x)). It's important to know that f(g(x)) does not usually have the same value as g(f(x)), so order matters when calculating their composition. A general rule to keep in mind is to work from the inside out: the innermost function is the one that you should start with.

EVALUATING FUNCTIONS

In this lesson I have learned about Evaluating a function means finding the value of f(x) =… or y =… that corresponds to a given value of x. To do this, simply replace all the x variables with whatever x has been assigned. For example, if we are asked to evaluate f(4), then x has been assigned the value of 4.
Example: Given that f(x) = 3x + 6, find f(2)

Solution: This means we will evaluate the function when x has been assigned the value of 2. The first step is to replace every x with 2. Then evaluate the function by following order of operations (BEDMAS).

f(x) = 3x + 6

So, we have :

f(2) = 3(2) + 6

        = 6 + 6

        = 12

In this lesson we talk about Rational functions have a polynomial in the nominator and denominator. Learn to define and write rational functions, putting together all of their elements, including vertical and horizontal asymptotes.

•RATIONAL FUNCTION is a function that is a fraction and has the property that both it's numerator and denominator are polynomials. In other words,R(x) is a rational function if R(x)=p(x)/q(x) where p(x) and q(x) are both polynomials.

Equations that contain rational expressions are called rational equations. For example, 
2x+1/4=x/3 is a rational equation. 
Some examples of rational functions are:
y=x+2/x−2;y=1/x²;y=x/x−3