Unizor - Math Concepts - Monotonic Functions скачать в хорошем качестве

Unizor - Math Concepts - Monotonic Functions

In the lecture associated with these notes we will only deal with monotonically increasing functions and omit the characteristic "increasing". All properties of monotonically increasing functions are trivially transformed into corresponding properties of monotonically decreasing functions. We now consider only functions defined on the domain of real numbers with real values. The reason for this is that we want to be able to compare different arguments and see how a change in an argument changes the value of a function. Let's assume further that with any increase of an argument, from any real number in a domain to any greater number also in a domain, the value of a function also increases. The functions that possess this property are called monotonically increasing. Similarly, if an increase in an argument results in decrease of a value of a function, this function is called monotonically decreasing. Both monotonically increasing and monotonically decreasing functions form a class of monotonic functions. Expressing these definitions in a more mathematical form, the function y = f(x) of real arguments and real values is called monotonically increasing if for any two arguments u, v that satisfy an inequality u is less than v the corresponding values of a function satisfy an inequality f(u) is less than f(v). Similarly, the function y = f(x) of real arguments and real values is called monotonically decreasing if for any two arguments u, v that satisfy an inequality u is less than v the corresponding values of a function satisfy an inequality f(u) is greater than f(v). As you see, we explicitly use a concept of comparison between arguments and values. That's why we have defined monotonic functions only among those that have real arguments and real values. More abstract functions like a color as a function of a flower or a function with complex arguments and values cannot be considered for this definition since there is no concept of "greater" or "less" for these elements. One-to-one Correspondence Monotonic function establishes one-to-one correspondence between its domain and its range. To prove it, we have to check that no two different arguments from its domain are mapped by this function into the same value from its range. Obviously, it's impossible for a monotonic function since two different arguments can only be in a relationship where the first one is greater than the second or it is less than the second. In each of these cases the values of a monotonic function will be different for these two arguments, greater or less than each other depending on whether the function monotonically increases or decreases. In any case the values will not be equal to each other. Monotonic and Inverse Since a monotonic function establishes one-to-one correspondence between its domain and its range, we consider an inverse function that takes as its arguments values from a range and takes as values the corresponding arguments of a main function. Thus, using this one-to-one correspondence, for any monotonic function we can define an inverse function. Inverse and Monotonic Though any monotic function has an inverse one, not every function that has an inverse one is monotonic. Consider a function defined on a segment from -1 to 1 with values y = -x + 1 for x from [-1,0) (not including x=0) y = x for x from [0,1] Clearly, it establishes a one-to-one correspondence between domain [-1, 1] and range [0, 2], but it's not monotonic, it decreases for negative arguments and increases for positive ones. Graph of an Inverse Function Graph of an inverse function is symmetrical to a graph of a main function relatively to an angle bisector between X-axis and Y-axis. Indeed, if a point (A, B) belongs to a graph of a main function y = f(x), an argument A is mapped by this function into value B, that is B = f(A). Inverse function y = g(x), by definition, establishes a correspondence between B as an argument and A as a function value, that is A = g(B). So, point (B, A) belongs to a graph of an inverse function y = g(x). Obviously, points (A, B) and (B, A) are symmetrical relatively to an angle bisector between X-axis and Y-axis. Therefore, for any point on a graph of a main function there is a corresponding point on a graph of an inverse function symmetrical relatively to a bisector between X-axis and Y-axis. That proves that entire graphs of a main and inverse functions are symmetrical relatively to this bisector.