Chapter 1: Differentiation; Section 2: Real Functions; page 2

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Classification by shapes

We can associate with each function a characteristic graphical display. On the other hand there are some characteristics that are common for group of functions, which allows to do some useful classifications.

Monotonic functions

A function is called monotonic increasing in an interval, if for any

 ${x}_{1}\text{\hspace{0.28em}}\text{and}\text{\hspace{0.28em}}\text{}{x}_{2}\text{ }\text{that}\text{\hspace{0.28em}}\text{}{x}_{1}<{x}_{2}\text{ }\text{holds:}\text{\hspace{0.28em}}\text{}f\left({x}_{1}\right)\le f\left({x}_{2}\right)$ (1.2.2.1)

The function is strictly increasing in an interval, if for any

 ${x}_{1}\text{\hspace{0.28em}}\text{and}\text{\hspace{0.28em}}\text{}{x}_{2}\text{ }\text{that}\text{\hspace{0.28em}}\text{}{x}_{1}<{x}_{2}\text{ }\text{holds:}\text{\hspace{0.28em}}\text{}f\left({x}_{1}\right) (1.2.2.2)

As an example we can take   $f\left(x\right)={x}^{3}$  and since  ${x}_{1}^{3}<{x}_{2}^{3}\text{ }\text{for any}\text{ }{x}_{1}<{x}_{2}$ , the function is strictly increasing for all the domain of real numbers.

Similarly one defines decreasing functions .

As an example the function   $f\left(x\right)=-x$  is strictly decreasing , since any   ${x}_{1}<{x}_{2}$  yields $-{x}_{1}>-{x}_{2}$ .

Examples of strictly monotonic functions are displayed on Fig. Strictly monotonic functions.

Bounded functions

If there is a constant  M , such that  $f\left(x\right)\le M$  for all x in an interval, M  is called an upper bound of the function for this interval.

If a function has a value  $f\left({x}_{0}\right)$ , such that for all  x  in an interval  $f\left(x\right)\le f\left({x}_{0}\right)$ , then   $f\left({x}_{0}\right)$  is called the absolute maximum of the function for this interval. Instead of $f\left({x}_{0}\right)$  one can use also $\underset{x\to {x}_{0}}{\mathrm{lim}}f\left(x\right)$ , where   ${x}_{0}$  does not have to belong to the domain of the function. From the definitions it follows, that an absolute maximum is also an upper bound, but not the opposite.

Appropriate definition of  $f\left(x\right)\ge m$  and  $f\left(x\right)\ge f\left({x}_{0}\right)$  exist for lower bound (m) and absolute minimum.

Local maxima and minima

A function  f(x)  has a local maximum at point  $x={x}_{0}$  if  x0  belongs to an (open) interval  (a,b)  such that  $f\left(x\right)  for all  $x\ne {x}_{0}$ (in that interval).

An alternative definition would be that the function is strictly increasing from the left (meaning  $a) and strictly decreasing from the right (meaning  ${x}_{0}\le x).

In a similar way one defines local mimnimum at point  $x={x}_{0}$ . The alternative definition would be that the function is strictly decreasing from the left, and strictly increasing from the right.

For simplicity in the future, a local minimum(maximum) will be called just minimum(maximum) and the adjective "local" will be omitted.

An example of a hand drawn bounded function with minima and maxima is explained at Fig. Shapes.

Parity of a function

The notion of parity is applicable only to functions in which, if  x  belongs to the domain of the function, so does  −x .

A function  f  is called even if the following holds:   $f\left(-x\right)=f\left(x\right)$ . Examples:  x² , |x|  etc.

Similarly, a function  f  is called odd if  $f\left(-x\right)=-f\left(x\right)$ . Examples: x , $\frac{|x|}{x}$  etc.

From the definition it follows that:

• Parity is conserved under the sum (or subtraction) of functions with the same parity.
• Multiplication or division of two functions with the same parity, yields an even function.
• Multiplication or division of two functions with opposite parity, yields an odd function.

A function  f  whose parity is undefined (neither even and nor odd), can be written as sum of an even (${f}_{+}$) and an odd (${f}_{-}$) functions in the following way:

 $f\left(x\right)={f}_{+}\left(x\right)+{f}_{-}\left(x\right)\left\{\begin{array}{l}{f}_{+}\left(x\right)=\frac{f\left(x\right)+f\left(-x\right)}{2}\\ {f}_{-}\left(x\right)=\frac{f\left(x\right)-f\left(-x\right)}{2}\end{array}\right\}$ (1.2.2.3)

Roots of a function

Any point of a function  y=f(x)  at which  y=0 , is called a root of the function. The number of roots and their locations is a characterisic of a function. A function can be rootless, (e.g.  y = x²+1) or have any number (even infinite) of roots. As an example the function  y = x(x−2)  has two roots at  x = 0  and at  x = 2 .

Inverse functions

As we already know, a function defines a correspodence between any number from the domain of the (independent) variables to the domain of the dependent variables. The inverse function defines the dependence in the reversed order.

As it was discussed in the previous section, if a function is multiple-valued, one uses to define branches that are single-valued functions by themselves. In the case of inverting a single-valued function, the result can be multiple-valued and care should be taken to define the necessary branches.

If we denote a function by  f  then the inverse function will be denoted by  invf . In the literature very often a different notation is used, namely  ${f}^{-1}$ . We are not going to adapt this notation because of the ambiguity with  ${f}^{-1}=\frac{1}{f}$ .

The definition of the inverse function can be formulated as:

 $\text{inv}f\left[f\left(x\right)\right]=f\left[\text{inv}f\left(x\right)\right]=x$ (1.2.2.4)

Indeed, if we apply an operation on a variable (x) and on the result apply the inverse operation, by definition we have to reproduce the variable unchanged. However, one has to apply caution when using (1.2.2.4), since the domains of the function and the inverse could be different.

As an example of (1.2.2.4) for  $f\left(x\right)={x}^{2}$  there is  $\text{inv}f\left(x\right)=\sqrt{x}$  and one obtains:

 $\sqrt{\left({x}^{2}\right)}={\left(\sqrt{x}\right)}^{2}=x$ (1.2.2.5)

There is however a restriction: one cannot use a negative value of x because the square-root is not defined for negative values.

Any inverse function can be found, either by the use of the definition (the dependence between the variables), or from (1.2.2.4). Is it possible to obtain graphically the inverse function directly from the graphical display of the function? Just use the x-axis as ordinate (dependent variable of the inverse function) by changing its direction to be vertical up, and then the y axis will be horizontal as the abscissa should be. Not so simple : The new abscissa is pointing to the left instead to the right. The coordinate system obtained this way is left-handed instead of right-handed. Does a graphical way exists to transform a left-handed to a right-handed coordinate system?

A solution of this puzzle using rotation in 3 dimensions, is presented in Fig. 3 dimensional rotation .

A second solution using mirror reflection is shown in Fig. Mirror reflection.

Now we are ready to apply this transformation to the coordinate system, together with the function in order to obtain the inverse function graphically.

The first example, simplified by the use of the same scale for the abscissa and the ordinate, deals graphically with the case of   $f\left(x\right)={x}^{2}$  and   $\text{inv}f\left(x\right)=±\sqrt{x}$   by applying the two methods: Fig. Inverse quadratic .

This example shows that we can apply the transformation on the function alone, and keep the axes without any change, if we name  y  the ordinate and  x - the abscissa for both: the function and the inverse function. Another important consequence is that it is simpler to obtain the inversion of a function graphically by mirror reflection.

In the next example, the scales of the abscissa and the ordinate are different, and therefore they should be interchanged after the inversion. The mirror reflection method is used. As in the previous case, the direction of the mirror should be that of the exact geometrical diagonal between the coordinate axes (in order to invert correctly the axes). The cubic function   $f\left(x\right)={x}^{3}$  is used to obtain   $\text{inv}f\left(x\right)=\sqrt[3]{x}$  as shown in Fig. Inverse cubic .

In the examples for graphical inversion of functions, we used diagonals passing through the origin. This is not necessary, but it simplifies the rescaling of the coordinates after inversion.

The shape of a function can provide some information about the inverse function, without any calculations. Here are some obvious examples:

• If a function is strictly increasing (decreasing) so is the inverse.
• If a function has an interval with constant value, the inverse function is infinite-valued at a point.
• If a function has a (local) minimum or maximum, the inverse function is multiple-valued (at least double-valued) in at least one interval.

There are no simple rules for analytically expressing an inverse function. We know for instance the inverse function of   ${x}^{n}$   where  n  is a non negative integer power, but we do not have a rule that gives us the mathematical expression for the inverse of a polynomial.

A polynomial of order n is a function defined by the following linear combination:

 ${p}_{n}\left(x\right)={a}_{0}{x}^{n}+{a}_{1}{x}^{n-1}+....+{a}_{n-1}x+{a}_{n}=\sum _{k=0}^{n}{a}_{k}{x}^{n-k}$ (1.2.2.6)

with  n  being a non negative integer and the factors  a - constants. It is elementary to find the inverse of  $y={p}_{n}\left(x\right)$  when  n = 2 , but for   $n\ge 5$  the mathematicians have proved that there is no analytical solution. This simple example shows the power of the graphical presentation.

As matter of fact we already used in (1.1.4.11) a polynomial as an example of the rules for physical dimensions. We are now going to reproduce it in a different context:

Physics teaches us that the linear motion with constant acceleration is:

 $l={l}_{0}+{v}_{0}t+\frac{1}{2}a{t}^{2}$ (1.2.2.7)
where:
• l  is the position (length from an origin) after a time  t
• ${l}_{0}$  is the position at  t = 0
• ${v}_{0}$ is the velocity at  t = 0
• a  is the acceleration

In (1.2.2.7)  l  is function of  t . We can reverse the question and ask at what time the distance is  l . By rewriting (1.2.2.7) as

 $a{t}^{2}+2{v}_{0}t+2\left({l}_{0}-l\right)=0$ (1.2.2.8)

one obtains the solution of t :

 $t=\frac{1}{a}\left(-{v}_{0}±\sqrt{{v}_{0}^{2}+2a\left(l-{l}_{0}\right)}\right)$ (1.2.2.9)

As expected this solution preserves the physical dimensions. The function  t  is undefined for values of  l  corresponding to the square-root of a negative number (1.2.2.9), which means that there is no physical solution. If the function  t  is defined, it can be double-valued. Then each branch of the function corresponds to a particular physical requirement, and the choice of the branch is not arbitrary.

Exercises

Study the following functions according to the guidelines:

• What is its domain of continuity?
• If applicable: find its parity, or alternately write it as the sum of functions with opposite parities.
• What are its roots?
• If appropriate, what is the behaviour of the function at infinity?
• What are the points of discontinuity, and what is the one-sided limit there?
• Where is the function increasing or decreasing?
• Find if they are minima and maxima!
• Find (if there are) bounds, absolute minimum or maximum!
• Give an approximate graphical presentation of the function. Connect points with straight lines.
• Calculate the inverse function, and check if it is consistent with the graphical presentation of the function that you drew previously!

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