Hello its me Drifter Programming! Today we continue with **Mathematical Analysis**, getting into some more stuff about **Functions**. We will talk about **Monotony, Extrema** etc. and lastly also get into some **Categories **or Types of Functions. It would be good to check out our** last post** about Functions first. So, without further do let's get straight into it!

## Function Monotony:

Suppose a function f: A -> R and B a subset of R.

- f is
**increasing in B**, when for every x1, x2 in B with**x1 < x2**implies that**f(x1) <= f(x2)**.**f ↑** - f is
**strictly increasing in B**when for every x1. x2 in B with**x1 < x2**implies that**f(x1) < f(x2)**. - f is
**decreasing in B**, when for every x1, x2 in B with**x1 < x2**implies that**f(x1) >= f(x2)**. f↓ - f is
**strictly decreasing in B**when for every x1. x2 in B with**x1 < x2**implies that**f(x1) > f(x2)**. - f is called a
**(strictly) monotonic function**when it is (strictly) increasing or (strictly) decreasing.

With all that we have that:

- When f is increasing and decreasing in B at the same time, well that is is a constant function (f(x) = c)
- When f is strictly monotonic in B it is also bounded in B and 1-1 (1-1 doesn't mean it is also strictly monotonic tho)
- When f is strictly monotonic (increasing or decreasing) then the inverse function f^-1 is also strictly monotonic (increasing or decreasing)
- To find out the monotonicity we use the division of change λ = (f(x2) - f(x1)) / (x2 - x2).

- λ > 0 => f is strictly increasing
- λ >= 0 => f is increasing
- λ < 0 => f is strictly decreasing
- λ <= 0 => f is decreasing
- λ = 0 => f is constant

**Example:**

Suppose the function f(x) = a*x^3, with a != 0.

- When a > 0, f is strictly increasing
- When a < 0, f is strictly decreasing
- Function f is 1-1, cause it is strictly monotonic in R.

## Extrema:

Suppose function f: A -> R.

- when there is a
**x0 in A**so that for every x in A we have that**f(x) <= f(x0)**then x0 is called the**global maximum**of f. The position x0 is called the maximum value of f. - when there is a
**x0 in A**so that for every x in A we have that**f(x) >= f(x0)**then x0 is called the**global minimum**of f. The position x0 is called the minimum value of f. - the global maximum and minimum are called the
**global extrema**of f. - when there is a
**x0 in A**so that for every x in a**subdivision of A**we have that**f(x) <= f(x0)**then x0 is called a**local maximum**of f. - when there is a
**x0 in A**so that for every x in a**subdivision of A**we have that**f(x) >= f(x0)**then x0 is called a**local minimum**of f.

**Example:**

Suppose f: A -> R, with f(x) = x^2 - 4x + 4.

xo = 2 is the global minimum of f with f(2) = 0.

We know this, cause f is strictly decreasing in (-∞, 2] and strictly increasing in [2, +∞).

(We will get into how we find this out posts later on)

# Function Categories:

**Linear Function [f(x) = a*x + b]:**

- The Graph of f contains all the points in 2d space for which y = a*x + b and so is a line ε
- The line ε intersects with the x axis in the point A(-b/a, 0
- The line ε intersects with the y axis in the point B(0, b)
- a = tan(ω) = y2 - y1 / x2 - x1 is called the slope of f
- With a = 0 we have a constant function f(x) = b that is parallel to the x axis

**Parabola Function [f(x) = a*x^2 + b*x + c, where a!=0]:**

- The parabola is intersecting with the y axis in the point A(0, c)
- The parabola is intersecting with the x axis depending on the discriminant's sign (Δ = b^2 - 4*a*c)
- When Δ > 0 the parabola is intersecting in two points X1( (-b -root(Δ))/2*a ,0) and X2( (-b +root(Δ))/2*a ,0)
- When Δ = 0 the parabola is intersecting in one point X(-b/2*a, 0
- When Δ < 0 the parabola is not intersecting with the x axis.
- The vertex of the parabola is at the position C(-b/2*a, -Δ/4*a)
- The convexity of the parabola depends on the sign of a (or mostly called p)
- When a>0 then the parabola is concave (or surfaces curve inwards)
- When a<0 then the parabola is convex (or surfaces curve outwards)
- The curvature of the parabola depends on the sign and the value of a

Here you can see how the value of a affects the different parabola functions:

**Cubic** **Function [f(x) = a*x^3, where a!=0]:**

**Hyperbola Function [y = a/x, where a!=0]:**

**Exponential Function [f(x) = a^x, where a> 0]:**

- For a > 1 the function is strictly increasing
- For 0 < a < 1 the function is strictly decreasing
- For a = 1 the function is constant and equal to 1 (1^x = 1)
- Any exponential function is strictly monotonic

**Logarithmic function [f(x) = loga(x), where x, a > 0]:**

- loga(x) = y <=> x = a^y, where a is called the basis
- When a = 10 we have log10 that is called a common logarithm
- When a = e = 2,718... we have ln that is called the natural logarithm
- Any logarithmic function is strictly monotonic
- A logarithmic function is the inverse of an exponential function

Properties:

**Trigonometric Functions:**

- You can read about trigonometric functions here in
**wikipedia** - You should know sin(x), cos(x), tan(x), cot(x) and the arc's (inverse functions) of them
- Here a list of trigonometric identities/rules from which we will need some later on

**Hyperbolic (Trigonometric)Functions:**

- You can read about hyperbolic function here in
**wikipedia** - You should know the sinh(x), cosh(x), tanh(x) and coth(x) and the inverses of them.
- You can also check out some of the identities/relations/formulas, cause we will may need them posts later on.

And this is actually it and I hope you enjoyed it!

Next time we will get into Limits and Continuity...

Bye!