Functions

Definition

A function is an assignment

notation: $f:A\to B,x\Rightarrow f(x)$

domain: $A$ codomain: $B$ range of a function: $f(A)=\{f(x)|x\in A\}$

Examples

• Assign each number it’s square
  • A: $\mathbb{R}$
  • B: $\mathbb{R}$
  • range: [0; inf)
• cost function of units produced $C(x)=100\ast x\ast \sqrt{x}+500$
  • A: $\mathbb{N}$
  • B: $\mathbb{R}$
  • range: $\{100\ast x\ast \sqrt{x}+500|x\in \mathbb{N}\}$

Absolute Function

$\mathbb{R}\to \mathbb{R}\text{+0}$ $x\to |x|$

Linear (affine) functions

$f(x)=\{\mathbb{R}\to \mathbb{R};x\to ax+b\}$ $a$ … slope $b$ … intercept

Equilibrium Price

$P:D(P)=S(P)$ Demand … $D(P)=-10P+400$ Supply … $S(P)=10P$ $P$ … Price

$-10P+400=10P$ $P=20$

Polynomials

For dregree $n$: $f(x)=\{\mathbb{R}\to \mathbb{R};x\to a_n{x}^n+a_{n-1}{x}^{n-1}+...+a_0\}$

Degree 2 - parabola opens

• upwards if $a>0$
• downwards if $a<0$
• $a{x}^2+bx+c=a(x+\frac{b}{2a}{)}^2-\frac{b^2-4ac}{4a}$
• the coordinates of the point of the maximum or minimum: $(\frac{-b}{2a}|\frac{-b^2-4ac}{4a})$
• Discriminant: $D=b^2-4ac$
• 0-Points: $x_{1,2}=\frac{-b\pm \sqrt{D^2}}{2a}$
• after $x_{1,2}$ → $f(x)=-\frac{1}{2}(x-x_1)(x-x_2)$
• then $f(x)=x^2+(-x_1)x+(-x_2)$

Power functions

$f(x)=\{\mathbb{R}\to \mathbb{R};x\to A{x}^r\}$

If $r<0$, then $x$ may not be $0$.

Exponential functions

$f(t)=\{\mathbb{R}\to \mathbb{R};x\to A{a}^t\}$

special case (“the” exponential function):

• $f(x)=e^x$

Gaussian density function (Normal Distribution)

• Gaussian distribution

Logarithmic functions

$f(x)=\{\mathbb{R}\to \mathbb{R};x\to {\log }_{a}x\}$ $lo{g}_{10}=log$ $lo{g}_e=ln$

Used in doubling times or halflife (Halbwertszeit) $t={\log }_{a}2$

$\ln 1=0$ $\ln e=1$ $\ln 4=\mathrm{1.3...}$ $\ln -6$ = n.d.

$\ln xy=\ln x+\ln y$ #wontfix get rest from slides on canvas

$\text{exp}(\ln x^2-2\ln y)=\frac{x^2}{y^2}$

Rule of 70

Rule of 70

The doubling time in years is approximately $\frac{70}{i}$ … $i$ being the percentage of interest $i\in \{1,2,3,...\}$

i	doubling time	4x time	1024x time
1	70	140	700
2	35	70	350
3	22.3	44.6	226
4	17.5	35	175
5	14	28	140
6	11.7	23.4	117
7	10	20	100
8	8.75	17.5	87.5
9	7.8	15.6	78
10	7	14	70
15	4.7	9.4	47
20	3.5	7	35
25	2.8	5.6	28
30	2.3	4.6	23

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