Function Types

A function has only one output for each input.
The set of all inputs of a function is called the domain.
The set of all outputs is called the range
Every input must has exactly one output.

Independent and Dependent Variables

y = x2 + 2
x is called the input variable or independent variable (x-axis)
y is called the output variable or dependent variable (y-axis)
Whatever the type of correspondence between the two variables we are always interested in the dependent variable.

Vertical Line Test

Functions are best described by plotting the co-ordinates of the input and output variables onto a graph
Here is f(x) - x(2) + 2 displayed as a graph
SS

You can use the vertical line test to decide if a graph represents a functions or not
The graph represents a function if any vertical line drawn only touches the line at one point
These lines represent functions

Relations

These lines do not represent functions and are just called relations

Splines
These are where data points are connected by a series of curves of assumed shape

Monotonic

Parabola Function

Even Function

Inverse Function

If f(x) = x^2 + 2
then f-1(x) = sqr(x) - 2

The inverse of a function is the mirror reflection in the y = x line

Piecewise Function

A function is called piecewise if it is made up of several pieces. Each piece has its own equation.

Composite Function

A composite function is the calculation of two functions
f(x) = x(2) + 2
g(x) = 6x + 1

If we wanted to calculate g(x) first and then f(x) it is written as g(f(x))

You always calculate the function on the inside first.
f(g(x)) != g(f(x))
The order of the composite functions is very important

Gamma Function

This can be thought of as a generalization of the factorial function.

Rational Function

Impulse Function

Delta Function

Convex Function

Inflection Points

Asymptotes