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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

Davy College

Analysis of Functions

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Analyzing Different Functions

Polynomial Functions:image00.png

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  image01.pngimage10.png
  • 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.

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In the case of maximums and minimums for any case of the functions there would be always absolute maximums and minimums. Because there are not turning points the relative max and min don’t exist in this type of function.

As we can see in the graphs above, there is a discontinuity when the exponent of the function is negative; the type of discontinuity that is present is the asymptote. For the other cases the asymptote discontinuity is the only type of discontinuity present in the functions. So the asymptotic behavior is to the x and y axes, vertical and horizontal.

The ending behavior when the exponents are positive is to be positive infinite when x gets to negative and positive infinite. When the degree is negative the ending behavior changes to have an asymptotic behavior to the x axis when the x values go to negative infinity and the same when is goes to positive infinity.

Rational functions:

The domain in the nominator can be the set of all real numbers, but in the dominator it is different because in fractions the denominator can’t be equal to zero so here are some examples of the domains with different types of rational functions:image02.pngimage25.png


The increasing and decreasing intervals depends in the degree of the nominator and the denominator here are some examples:


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As we can see in the graphs above the increasing and decreasing intervals depends on the degree of the function, for example in the case of the linear ones, the sign of the slope would tell if the line is increasing or decreasing.

These functions can be injective, in the case of linear functions, but also bijective for degree 2 or more functions.

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

These type of functions are not periodic because in the formula f(x+a)=f(x), a is not constant, so it can’t be periodic.

There can be symmetry in some functions like the quadratic, linear, cubic, etc. but it’s important to mention that not always there would be symmetry among all these degrees, it depends on the quotients of the function. Similar as the polynomial functions the maximums can be absolute an relative, depending on the degree of the function. Is important to have in consideration that all these characteristics depend on the domain given before, because with the domain just a part of the function is considerate and not all, which can contradict what, was said before.

There can be discontinuity but the functions can be also continuous all depends on the degrees or form of the function but as well the domain given. So he ending behavior follows the same conditions.

Hybrid functions:

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