All rational functions have inversions


Requirements and preliminary remarks

In the first chapter of functions we got to know the following concepts: In this chapter we assume that you are familiar. Please repeat the relevant passages if necessary. (A first activation of one of the links above opens a new browser window with the first chapter of functions. If you leave it open, it will also be used by later calls without further loading time).      

Features 1

We will mainly focus on real functions speak, i.e. of functions that either work for all real numbers or for a subset of R. are defined, i.e. functions of the type f : R.®R. or f : A.®R. With A. ÍR.. Below we will discuss examples of functions defined on other sets.

An important topic in this chapter is Properties of functions With Properties of their graphs to bring in context. Some function graphs are shown in the form of prepared graphics. Please do not hesitate to use the whenever you also want to look at the graph of a (term-defined) function. (If there are problems with the ^ symbol for entering powers in your browser, replace it with the word up).

The definition of a function f can about the Assignment rule  f : x®f (x) or in the form of the Function equation  y  =  f (x) happen. The designation of the "independent" and the "dependent" variables with the letters x and y is not absolutely necessary, but an old tradition. Accordingly, when discussing graphs, we will refer to the "horizontal axis" ("abscissa") as x-Axis and the "vertical" axis ("ordinate") as y-Axis describe.

Finally, one more term that will appear more often in the following: A rational function is a function that can be written as the quotient of two polynomials. Rational functions are on the one hand manageable and relatively easy to study, on the other hand diverse enough to play an important role in mathematics lessons.

If you're just interested in some of the concepts discussed below, so do skip The others or skim them roughly.


real numbers
Symmetry and antisymmetry

We call a function f : R.®R.  symmetrical (sometimes too just) if for everyone xÎR.
applies. The graph of a symmetric function is symmetric with respect to the y-Axis (i.e. it goes under a reflection on the y-Axis in itself above). We also name a function f : R.®R.  antisymmetric (sometimes too odd) if for everyone xÎR.
applies. The graph of an antisymmetric function is symmetric with respect to the origin (i.e. it turns into itself under a point reflection at the origin, i.e. a rotation by 180 °). Both concepts are casually carried over to functions that are not entirely R. are defined: It must then (1) or (2) for all xÎA. apply, where A. the domain of f is. Many important functions fall into one of the two classes:           
  • Examples of symmetric functions: xn for straight n (i.e. 1, x2, x4,...), x2-1, 1/x2, 1/x4, 1/(x2-1), (1-x2)1/2, cosx, sin2x, x sinx, coshx (for the latter see below).
  • Examples of antisymmetric functions: xn for odd n (i.e. x, x3,...), x3-x, 1/x, 1/x3, x/(x2-1), x (1-x2)1/2, sinx, x cosx, tanx, cotx, sinhx, tanhx, cothx (for the last three see below), so-calledx (see below).
As can be seen from these examples, the (anti-) symmetry of many elementary functions of simple identity (-x)2 = x2 owe, which in turn is a consequence of (-1)2 = 1 is. Therefore defines a function term in which the variable x only square (i.e. as x2) always has a symmetrical function. The product of two symmetric or two antisymmetric functions is symmetric, the product of a symmetric and an antisymmetric function is antisymmetric.

Symmetry properties of functions can be used to keep calculations as short as possible: Is a property of an (anti) symmetrical function (e.g. the course of its graph or the position of a zero) in the range x If ³ 0 is known, the corresponding property results for the area x <0 completely automatically.


We call a function f : R.®R.  periodicallyif it's a positive number p there so that for everyone xÎR.
applies. p is called then period or Period length. With growing x A periodic function "repeats" over and over again, because applying (3) twice results f (x + 2 p)  =  f (x + p)  = f (x), and in general applies f (x + n p)  =  f (x) for every natural number. With p is therefore also every multiple n p a period, and accordingly the graph of a periodic function has a repeating pattern. The concept casually carries over to functions that are not entirely R. are defined: It must then (1) for all xÎA. apply, where A. the domain of f is.

Some periodic functions have a smallest Period. (We call p the smallest period when p Period is, but any number q with 0 <q < p is not a period). This is usually given to characterize periodic functions (without always mentioning the addition "smallest"). The remark opposite shows that not every periodic function has a smallest period.
            Examples of periodic functions:
functionSmallest period
sinx, cosx 2p
tanx, cotx, sin2x, cos2xp
Sawtooth function: In the interval -1 £x £ 1 will

f (x)  =  1 - |x|

is defined outside f continued periodically.

As the last example shows, it is very easy to get through periodic continuation define many more periodic functions.

Periodicity properties of functions can be used to keep calculations as short as possible: For example, it is of a function with a period p a property (e.g. the course of your graph or the position of a zero) in the range 0 £x < p known, the corresponding property results for all others x completely automatically.

Periodic functions are needed to Vibration processes to model, being then x stands for time and p as Period duration referred to as. One method of analyzing periodic functions more precisely (e.g. to investigate what the timbre of a musical instrument physically consists of) is that Fourier analysiswhich will be discussed in a later chapter.


Fourier analysis


We only mention this term superficially here, as a separate chapter is devoted to its deeper aspects. One at a time A. defined function f : A.®R. is called steadily when small changes are made to x within A. small changes from f (x) have as a consequence. The Graph of a continuous function is a contiguous curve (which can be traced with a pencil, so to speak, without removing it). The concept of continuity only makes sense for intervals in which a function is defined. Is a function in several Defined intervals (such as 1 /xwhat yes for x = 0 does not exist), each of these areas (for 1 /x are these the two intervals x <0 and x > 0) must be considered separately. A discontinuous Function is characterized by the requirement for a coherent graph in the domain of definition is not fulfilled, that is, for example, a Jump point exists (on which the function is defined, but on which the graph is "torn apart"). A function that is discontinuous at isolated points, but continuous in between, is called piece by piece (or in sections) steadily. But there are also functions that work on quite R. defined and on everyone Position are discontinuous.

examples for (U.N)continuous functions:


The functionis steady... and...
x2 in whole R.  
|x| in whole R.  
1/x in the two intervals
x <0 and x > 0
is not defined for x = 0
x/(x2-1) in the three intervals
x < -1, -1 < x <1 and x > 1
is not defined for x = -1 and x = 1
x1/2 in the interval x ³ 0 is not defined for x < 0
in the two intervals
x <2 and x > 2
is discontinuous at x = 2
(Jump point)
It is Q the set of rational numbers. nowhere is for everyone x ÎR. discontinuous

rational numbers

Only the last two examples represent discontinuous functions, the others are continuous (in their respective domains).

A function that is described by a term that can be built up using the basic types of calculation from powers, trigonometric functions and their inverses, exponential functions and logarithms is continuous in its domain of definition. In this sense are term-defined functions always steadily. This is especially true for most of the functions discussed in this chapter. However, below we will also see some useful discontinuous functions.


zeropoint and their relationship to Equations we got to know it in the first chapter on functions. The simplest functions whose zeros are worth investigating are the Polynomials, which we will discuss in more detail below. Polynomials can behave very differently in the vicinity of zeros: x0 called Zeron-th order the polynomial function f, if
It is n a natural number and c a non-zero constant.

      The behavior of f Near x0 resembles that of the power function (of the "Monoms") cxn near 0. The greater the order n is, the faster the function value falls to zero when x approaches the zero, and all the more so flatter is the graph near them. Is n straight so has function on both sides of x0 the same sign (as it is for the function x2 the case is). Is n odd, the signs are on both sides of x0 different (as for the function x), and the graph "crosses" the x-Axis.

We'll show below when it comes to polynomials that each Is of this kind, i.e. has a well-defined order.
             The concept of the ordering of a zero can be extended to a larger class of functions. For example, has the function f (x) = sin2(x) a second order zero for every integer multiple of p. However, there are also functions whose zeros do not fit into this scheme (example: the absolute value function f (x) = |x|).


Order of zeros, in general
Singularities and poles

If a real function is given by a term, it is initially not necessarily for all xÎR. well defined. Several things can happen in the process.
  • It can happen that a term is not well defined at a certain point, but can be made into a continuous function by a subsequent definition of the missing function value. We then speak of one Definition gap, the "steadily closed"can be, or one liftable singularity. You can call up an example with the help of the adjacent button.
  • But it can also happen that a term has a Infinity point (Singularity) owns. The simplest example is 1 /x. There's nothing to shake here - the closer x the number 0 comes, the greater the amount of 1 /x. The spot x = 0 definitely does not belong to the domain of definition (that is, as A. = R. \ {0} can be accepted).

    This begs the question of how a function behaves near an infinity point (in particular, "how fast" it "goes to infinity"). For rational functions, i.e. quotients of two polynomials (see below), this question can be answered systematically. Is x0 an infinity point of the rational function fso their behavior takes near x0 the shape, where n a natural number and k is a non-zero constant. x0 is called pole (Pole position) n-th order designated. To get an idea why this is and how to do it n and k determined for a given rational function, click on the adjacent button.

    The behavior near a pole is thus based on the (relatively simple) functions of the form 1 / (x-x0) n measured. The bigger n is, the more rapidly the amount of the function increases when x the pole x0 approaching. Is n even, the function has the same sign on both sides of the pole (as it is for the function 1 /x2 is the case), and the graph shows two "branches", which at the pole point either both extend upwards or both extend downwards "to infinity". Is n odd, the signs are different on both sides of the pole (as for the function 1 /x), and the graph looks torn: one "branch" rises to "positive infinite", while the other falls to "negative infinite".

    The Order of a pole can with the Order of a zero can be related to: the faster f near a pole increases, the faster falls 1/f there towards zero. If we compare (5) with (4), we get: f has at x0 a pole n-th order if 1 /f there a zero n-ter order possesses.
    The concept of the pole can be extended to a larger class of functions. For example, the tangent function has a pole of the first order at the point p / 2.
  • Completely different things can also happen, which, however, cannot easily be put into a uniform scheme and are best analyzed when they occur.
    An example: 1 /x + 1/|x|. Look at the graph by entering 1 / x + 1 / abs (x) into the function plotter and try to understand it based on the term that defines it! (Tip: Calculate the function term separately for x <0 and x > 0!)
    Another example: sin (1 /x) - also take a look at their graphs!

Pol, in general
Asymptotes and asymptotic behavior

Does the graph one function tends to be one Straight lines getting closer and closer, so will this asymptote called. Asymptotes occur
  • when the behavior of a function for large values ​​of x (or -x) the one linear function becomes more and more similar and
  • at Infinity places.
In the first case we are talking about the asymptotic approximation a function (a graph) to a linear function (a straight line). To express the idea that about the behavior of a function for large values ​​of the independent variable that increase beyond any bound x is spoken, formulations such as
  • "For x® "(pronounced" for x towards infinity ") or
  • "for big x"

      used. Specifically intended to have a behavior for under every bound falling values ​​of x (i.e. above every barrier growing -x) are spoken, the phrase "stands forx®- "(pronounced" for x towards minus infinity "). It depends on the sign of x not to, it is sometimes in the form "for large |x| "or" for |x| ® ". There is also the casual phrase" in infinity "or simply the word" asymptotic ". How the ideas behind it can be formulated more precisely is the subject of other chapters.

® ¥
     Danger: The arrow ® in a formulation like "for x® "has nothing to do with the arrow in the assignment rules for functions! These two types of arrows should not be confused!

example: How does it behave f (x) = (2x2- 3x + 4)/x defined function for large x? We reform and rewrite the functional term as 2x- 3 + 4/x. The bigger x is, the smaller the last post is. Hence approaching f (x) For x® the values ​​of the linear function G(x) = 2x- 3. Its graph is a straight line and represents an asymptote of f further has f a pole at x = 0. Since the graph is the yAxis, this is also an asymptote of f.

To learn how to get the asymptotes of a given rational function, i.e. a quotient of two polynomials (see below) systematically determined, click on the adjacent button.

Not only rational functions can have asymptotes. For example, this is represented by the equation y = 2x defined straight line an asymptote of the function H(x) = 2x + e-xas this is for x® the behavior of the linear function k(x) = 2x accepts. The idea of ​​"asymptotic behavior"can be generalized and refined even further: For example, it can be said that (2 + e-x)/x2 for big x the same asymptotic behavior as 2 /x2 has, or that x2 + 1/x at infinity the behavior of x2 accepts. (Use the function plotter to see the graphs of the examples given here). The purpose of such statements is on the one hand to get a grip on the "global" behavior of functions, and on the other hand to use simple and well-known functions as a yardstick for the behavior of more complex functions.


Equations of lines
Convexity behavior

To put it bluntly, let's call a function f : R.®R.  convex (or "open at the top") if any connection between two points on the graph of f is at no point "below" this graph. A function is called analog f : R.®R.  concave ("open at the bottom") if any connection between two points on the graph of f is at no point "above" this graph. The terms "below" and "above" refer to the coordinate system in which the graph of a function is viewed: one point is "above" another if its yCoordinate is larger. (Danger: In the literature, the two terms can sometimes be found with interchanged meanings!)

A function is convex if the negative function to it is concave (and vice versa).

These two terms can also be applied to functions that are not entirely R., are defined, although they only make sense if they are related to an interval (i.e. a coherent area). A function can have different convexity behavior in different intervals of its domain.

  • Examples of convex functions: x2, x4, x3 in the area x  0, 1/x in the area x > 0, ex, e-x, |x|.
  • Examples of concave functions: -x2, x1/2 (in its domain of definition x  0), x3 in the area x  0, 1/x in the area x <0, lnx (in its domain of definition x > 0).

We call a function f : A.®R.  limited upwardsif there is a number c with the property
gives. c is called then upper bound. Analog means flimited downwardsif there is a number k with the property
gives. k is called then lower bound. A feature that goes up and is restricted below, is used without further specification as limited (and a function that is not limited as a unlimited). The graph of a function constrained upwards (downwards) is always below (above) a for x-Axis parallel straight lines. Some examples:           
  • Limited to the top: -x2, 1 - x4.
  • Limited downwards: x2, x4 - 3, ex, |x|.
  • Up and limited below: 1/(x2 + 1), sinx, cosx.
We will take up this concept again in a later chapter and ask, for example, when there is a smallest upper bound and a largest lower bound.

Combine functions

There are numerous ways of combining functions with one another:
  • Sum, difference, product, quotient: What does it actually mean to "add two functions"? Is A. ÍR. and are f : A.®R. and G : A.®R. two real functions with (the same) domain they can have a third function f + g : A.®R. (also with definition area A.) by
    To be defined. It is said that the sum of two functions point by point (i.e. for each "point" x as the sum of the function values ​​of f and G) is defined. The difference and the product of two functions can be defined in a completely analogous way. This way the set of all functions A.®R. equipped with the "arithmetic operations" plus, minus and times. Hence the square f 2 a function f are formed. (In this sense it is also common to sin2x for (sinx)2 to write). We have to be a little careful with the quotient, since we are not allowed to divide by zero. He can point by point as a function f/G : B.®R. can be defined, where B. = { xÎA. | G(x) 0} is.
  • Concatenation (Execute one after the other): What is the structure of the term sin (x2) defined function? You will be on a x applied by first on x the function "squaring" and after that on the result (i.e. x2) the "sine" function is used. The same can also be done with other functions: Only the function becomes G on x applied and then the function on the result fso we get f (G(x)). The function that combines these two steps is called Concatenation (sometimes too shortcut) of f and G and will with f OG designated:
    The range of values ​​from G a subset of the domain of f otherwise it would be a x there, for that though G(x), no but f (G(x)) is not defined.

    Within the set of all functions R.®R. can be chained as required. Concatenation is an "operation" which, like multiplication, turns two functions into a third, and like multiplication, it fulfills the so-called "Associative law"
    However, it depends on the order. f OG is not the same as G Of. (Example: sin2(x) is not the same as sin (x2)). In mathematical terminology this means: The concatenation is Not "commutative". The order in which the two functions are used runs a bit against our intuition: despite the name"f OG" becomes firstG and after thatf applied. (The danger of confusing the order arises from the fact that in the usual notation "f (x) "the function icon fLeft