# Basic Functions

## Functions

Basic Functions are real-valued functions of a real variable. Basic functions can be represented by an analytical expression. At the elementary functions include rational functions, trigonometric functions, exponential and logarithmic functions and their inverse functions.

In general, a function is a mapping between two quantities that accurately assigns to each element of one set is an element of the other.

### Definition

A function f assigns to each element of a quantitative definition D exactly one element of the target set Z. Assigned for the only the element x ∈ D element of the target set to write in general f (x).

$f\phantom{\rule{0.3em}{0ex}}:\phantom{\rule{0.3em}{0ex}}D\phantom{\rule{0.3em}{0ex}}\to \phantom{\rule{0.3em}{0ex}}Z\phantom{\rule{0.3em}{0ex}},\phantom{\rule{0.3em}{0ex}}x\phantom{\rule{0.3em}{0ex}}↦\phantom{\rule{0.3em}{0ex}}y$

### Analytical properties

##### Monoton

In mathematics, ie a function or sequence that is always greater with increasing function argument or constant (ie never falls), monotonically increasing (or monotonically increasing or isotonic). Accordingly, ie a function or sequence monotonically decreasing (antitone) when it is only smaller or remains constant. The values ​​of the function or the terms of the sequence change anywhere, it is called constant. Strictly increasing (resp. strictly decreasing) are functions or sequences that are only larger (smaller), but are nowhere constant.

##### Continuous

A function is called continuous if sufficiently small changes of the argument (argument) to arbitrarily small changes in the function value. This means in particular that no jumps occur in the function values​​.

##### Bounded

A function is called bounded if the function values ​​f(x) does not exceed an upper or lower bound. That if f(x) ≤ M for all real numbers x, then the function f is bounded from above. When M is the smallest integer for which holds, then M is the upper limit of the function f with respect to the analogous definition applies to the lower limit. F(x) for all x is greater than m, then m is the lower limit.

## Rational Functions

### Linear Functions

x-min= x-max=

$f\left(x\right)=a·x+b$

$\text{mit}\phantom{\rule{1em}{0ex}}a,b\phantom{\rule{0.5em}{0ex}}\in \phantom{\rule{0.5em}{0ex}}\mathbb{R}\phantom{\rule{1em}{0ex}}\text{und}\phantom{\rule{1em}{0ex}}a\phantom{\rule{0.5em}{0ex}}\ne \phantom{\rule{0.5em}{0ex}}0$

a= b=

#### Properties

Linear functions are rational integral functions. They grow monotonically for a> 0 and fall monotonically for a <0 Images of linear functions are straight lines.

x-min= x-max=

#### Properties

Quadratic functions are rational integral functions of the second degree. The image of a quadratic function is a parabola that opens upward or downward. For a> 0, the parabola opens upward, for a <0 down.

##### Vertex S

$S\left(-\frac{b}{2a},c-\frac{{c}^{2}}{4a}\right)$

##### Derivative f'

$f\prime \left(x\right)=2·a·x+b$

##### Product representation

$f\left(x\right)=\left(x-{x}_{1}\right)\left(x-{x}_{2}\right)$

$f\left(x\right)=a·{x}^{2}+b·x+c$

$\text{mit}\phantom{\rule{1em}{0ex}}a,b,c\phantom{\rule{0.5em}{0ex}}\in \phantom{\rule{0.5em}{0ex}}\mathbb{R}\phantom{\rule{1em}{0ex}}\text{und}\phantom{\rule{1em}{0ex}}a\phantom{\rule{0.5em}{0ex}}\ne \phantom{\rule{0.5em}{0ex}}0$

a= b= c=

## Trigonometric Functions

### General sine function

x-min= x-max=

In unfavorable selected interval in terms of the frequency may arise due to a false picture of the function by the resolution of the display. The maxima and minima of the function are always constant. In case of deviations, the interval has to be chosen smaller.

#### Properties

For a = b = 1 and c = 0, the ordinary sine function f (x) = sin (x) is available. It is a periodic function with period 2 π and the value range -1 ≤ f (x) ≤ 1

The general sine function is shown by an affine transformation from the ordinary sine function. Stretching by a factor a in the direction of the y-axis, with the elongation factor 1 / b in the x-axis followed by parallel displacement in the direction of the x-axis to c / b.

Often, the parameters of the general sine function are named as follows:

$f\left(x\right)=A\mathrm{sin}\left(\omega x+{\phi }_{0}\right)$

here A denotes the amplitude, ω is the angular frequency and φ 0 , the initial phase.

Many processes in nature and technology that reflect the character of oscillations, can be described by the general sine function.

If the variable is of the general sine function, the time t then it is called a harmonic oscillation:

$f\left(t\right)=A\mathrm{sin}\left(\omega t+{\phi }_{0}\right)$

the phase shift φ0 of the amplitude A, the period T = 2 π / ω and the frequency f = 1 / T.

$f\left(x\right)=a\mathrm{sin}\left(bx+c\right)$

$\text{mit}\phantom{\rule{1em}{0ex}}a,b,c\phantom{\rule{0.5em}{0ex}}\in \phantom{\rule{0.5em}{0ex}}\mathbb{R}\phantom{\rule{1em}{0ex}}\text{und}\phantom{\rule{1em}{0ex}}a,b\phantom{\rule{0.5em}{0ex}}>\phantom{\rule{0.5em}{0ex}}0$

a= b= c=

## Exponential Functions

### Exponential function exp(x)

x-min= x-max=

#### Properties

The exponential function is defined for all real numbers. The function has no zeros and no extremes. The function values ​​are always positive and the function is for b> 0 monotonically increasing and monotonically decreasing for b <0. The natural exponential function e x is available for b = 1

the addition theorem applies to the exponential function:

$f\left({x}_{1}+{x}_{2}\right)=f\left({x}_{1}\right)+f\left({x}_{2}\right)$

The Inverse of the Exponential function f(x) = ax is the Logarithm:

${f}^{-1}\left(x\right)={\mathrm{log}}_{a}\left(x\right)$

In the special case f(x) = ex is the Inverse the Logaritm naturalis:

${f}^{-1}\left(x\right)={\mathrm{log}}_{e}\left(x\right)=\mathrm{ln}\left(x\right)$

#### Applications

Description of organic growth: with g 0 initial size and growth of the constants c

$g\left(t\right)={g}_{0}\phantom{\rule{0.3em}{0ex}}{e}^{ct}$

Description of a process of decay: with m0 initial size and λ the decay constants

$m\left(t\right)={m}_{0}\phantom{\rule{0.3em}{0ex}}{e}^{-\lambda t}$

Damped Vibration: with the damping constant R

$f\left(t\right)={e}^{-Rt}\mathrm{sin}\left(\omega t+{\phi }_{0}\right)$

$f\left(x\right)=c\phantom{\rule{0.3em}{0ex}}{e}^{bx}$

$\text{mit}\phantom{\rule{1em}{0ex}}b,c\phantom{\rule{0.5em}{0ex}}\in \phantom{\rule{0.5em}{0ex}}\mathbb{R}\phantom{\rule{1em}{0ex}}\text{und}\phantom{\rule{1em}{0ex}}b,c\phantom{\rule{0.5em}{0ex}}\ne \phantom{\rule{0.5em}{0ex}}0$

$\text{Setzt man}\phantom{\rule{1em}{0ex}}a=c{e}^{b}\phantom{\rule{0.5em}{0ex}}\text{so gilt:}$

$f\left(x\right)=c\phantom{\rule{0.3em}{0ex}}{e}^{bx}={a}^{x}$

b= c=

### Error function exp(-x2)

x-min= x-max=

#### Properties

The function x is defined for each real value and takes its maximum at x = 0 with y = b to. The image of the function is symmetrical about the y-axis.

Derivative:

$f\prime \left(x\right)=-2{a}^{2}bx\phantom{\rule{0.3em}{0ex}}{e}^{-{\left(ax\right)}^{2}}$

Turning points:

${x}_{\mathrm{w1}}=\frac{1}{a\sqrt{2}}\phantom{\rule{0.5em}{0ex}};\phantom{\rule{0.5em}{0ex}}{y}_{\mathrm{w1}}=\frac{b}{\sqrt{e}}$

${x}_{\mathrm{w2}}=-\frac{1}{a\sqrt{2}}\phantom{\rule{0.5em}{0ex}};\phantom{\rule{0.5em}{0ex}}{y}_{\mathrm{w2}}=\frac{b}{\sqrt{e}}$

${f\prime }_{\mathrm{w1}}=ab\sqrt{\frac{2}{e}}$

${f\prime }_{\mathrm{w2}}=-ab\sqrt{\frac{2}{e}}$

#### Special case

For

$b=\frac{1}{\sqrt{2\pi }}\phantom{\rule{0.5em}{0ex}};\phantom{\rule{0.5em}{0ex}}a=\frac{1}{\sqrt{2}}$

there is the bell curve. Describes the probability density of the normal distribution.

$f\left(x\right)=b\phantom{\rule{0.3em}{0ex}}{e}^{-{\left(ax\right)}^{2}}$

$\text{mit}\phantom{\rule{1em}{0ex}}b\phantom{\rule{0.5em}{0ex}}>\phantom{\rule{0.5em}{0ex}}0\phantom{\rule{1em}{0ex}}\text{und}\phantom{\rule{1em}{0ex}}a\phantom{\rule{0.5em}{0ex}}\ne \phantom{\rule{0.5em}{0ex}}0$

a= b=

### Normal distribution (Gaussian distribution)

x-min= x-max=

#### Properties

The image of the Gaussian distribution is a bell-shaped curve with the maximum and the symetric center at a. The σ parameter defines the spacing of the turning points from symetric center. If σ small, the bell curve is narrow and pointed. For large σ function is flat and wide.

$f\left(x\right)=\frac{1}{\sqrt{2\pi }\sigma }\phantom{\rule{0.3em}{0ex}}{e}^{-\frac{1}{2}\frac{{\left(x-a\right)}^{2}}{{\sigma }^{2}}}$

$\text{with}\phantom{\rule{1em}{0ex}}\sigma \phantom{\rule{0.5em}{0ex}}>\phantom{\rule{0.5em}{0ex}}0\phantom{\rule{1em}{0ex}}\text{und}\phantom{\rule{1em}{0ex}}a\phantom{\rule{0.5em}{0ex}}\in \phantom{\rule{0.5em}{0ex}}\mathbb{R}$

a= σ=