Zeroes of Functions

A zero of a function f is a number c for which f(c)=0. A zero c of a function f can be a real or a complex number. A complex number is a number of the form z=a+bi, where I squared equals -1 and a and b are real numbers. The number a is called the real part of z and b which is called the imaginary part of z. The symbol I is called the imaginary unit and it is common practice to write it as I=square root of -1. If z=a+bi is a complex number, then z=a-bi is called its conjugate. Thus the simple polynomial function f(x)=x squared plus 1 equals zero are plus/minus square root of negative one, that is , I and -I. 

Here is a picture of a problem that involves zereos of a function! 

Piecewise- Defined Functions

A function f may involve two or more expressions or formulas, with each formula defined on different parts of the domain of f. A function defined in this manner is called a piecewise- defined function. Piecewise functions are written using the common functional notation, where the main part of the function is an array of functions and associated subdomains. Most importantly, there must be a finite number of subdomains, each of which must be an interval, in order for the overall function to be called "piecewise". For absolute value piecewise functions all values of x less than zero, the first function (−x) is used. Which cancels the sign of the input value, making negative numbers positive. For all values of x greater than or equal to zero, the second function (x) is used, which evaluates trivially to the input value itself.

 

Here is a picture of a piecewise function being graphed. 

The F(x) Men- Translations and Transformations

Explanation:

In translations there are vertical and horitizontal shifts, depending on wether the number is inside parenthesis or outside the parenthesis. There is also a rigid transformation, which is a transformation that changes only the position of the graph in the xy- plane but not it's shape. One way of rigidly transforming a graph of a function is by a reflection in a coordinate axis. If y=f(x) is a function, then the graph of y=-f(x) is the graph of f reflected in the x-axis and y=f(-x) is the graph of f reflected in the y-axis. A nonrigid transformation is when the shape of the graph is changed but retains, roughly, it's original shape. Some examples of nonrigid transformations are stretching or compressing. A graph is vertically stretched by a factor of c units if c> 1, and vertically compressed by a factor of c units if 0<c<1.