Function

A function is a mathematical object that takes an input and produces an output (number, vector, etc).

Hyperbolic functions are sinus, cosinus, tangente

Notation

$$f:X^n\to Y$$

Function $f$ from domain $X$ with $n$ variables to codomain $Y$. Note that there is multiple example of different types of domains and codomains. Find more here.

Convex / concave

There is two methods two know if a function is convex:

• If we trace a line between two points from the function and that it's on or above the function (in the interval between those two points)
• If the 2nd derivative is positive (on a given interval) $f(x)=x^2$ is convex as $f^{\prime }(x)=2x$, $f^{″}(x)=2>0$

A function $f(x)$ is concave when its opposite $-f(x)$ is convex.