Analysis of Functions. The factors of decreasing and decreasing intervals (in the y axis) in a polynomial function depend on the turning points of the function.

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Gerald Veliz


Analyzing Different Functions

Polynomial Functions:

These are functions with x as an input variable, made up of several terms, each term is made up of two factors, the first being a real number coefficient, and the second being x raised to some non-negative integer exponent.

The domain of any polynomial functions are the real numbers set, R. These are some examples with different degrees:

  • f(x) = x²  

  • f(x) = x3 

        

The factors of decreasing and decreasing intervals (in the y axis) in a polynomial function depend on the turning points of the function. The values of y can first be increasing but suddenly caused by the turning point, the values start to decrease. For example the values of y increase on the open intervals (∞ < x < a) but then when b < x < ∞ the intervals start decreasing.

Even polynomial functions have f(x) = f(-x) and odd polynomial functions occur when f(-x)= -f(x)Polynomial functions can be even, odd or neither, as shown with the following examples:

        

A polynomial function is not periodic in general because a periodic function repeats function values after regular intervals. It is defined as a function for which f(x+a) = f(x), where T is the period of the function. In the case of polynomial functions, clearly the only exception that can’t be a periodic function is that there is no definite or constant period like "a". The relation of periodicity, however, holds for any change to x, so it can also be accepted the idea that polynomial functions are periodic functions with no period.

There are two types of maximums and minimums, which are relative and absolute. In polynomial functions there can be both of them, but always just one absolute maximum and one minimum (if the degree of the function is odd), or two maximums or minimums (if the degree is even).the amount of relative max and min depends on the number of tuning points, as there are more turning points, there would be more relatives.

As we can easily observe in the graph of a quadratic function is that with this degree the function is subjective because there is more than one matching, meaning that there are two numbers in set A which has the same number in set B. the same occurs with the functions with degree 4. But in the case of cubic and degree 5 functions the relation can be injective and surjective because they can have a one to one relation but also a many to one so depending on the quotients of the functions.

Power Function:

If p is a non-zero integer, then the domain of the power function f(x) = kxp consists of all real numbers. For rational exponents p, xp is always defined for positive x, but we cannot extract an even root of a negative number. Any rational number p can be written in the form p = r/s where all common factors of r and s have been cancelled. When this has been done, kxp has domain:        

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  • All real numbers if s is odd
  • All non-negative real numbers if s is even.

If p is a real number which is not rational (called an irrational number), then the domain of xp consists of all non-negative real numbers.

In power functions when the degree is positive then there would be always increasing values doesn’t matter if it’s odd or even, but when the degree is negative then it depends if it’s odd or even, to have increasing or decreasing intervals.

In power functions when the degree are positive integers then the function is ...

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