1.1 Four Ways to represent a Function

1. Piecewise Defined Functions

1.1. - Piecewise defined function : A function that is defined by different formulas in different parts of their domain.

1.2. - The absolute value funcion is an example of of a piecewise defined function.

2. Symmetry

2.1. - If a function satisfies f(-x)=f(x) for every number x in its domain, then f is called an even function.

2.2. - The graph of an even function is symmetric about the y-axis.

2.3. - if a function satisfies f(-x)=-f(x) for every number x in its domain, then f is called an odd function.

2.4. - The graph of an odd function is symmetric about the origin.

3. Increasing and Decreasing Functions

3.1. - A function f is called increasing on an interval (I) if f(x1) < f(x2) whenever x1<x2.

3.2. - A function is called decreasing on an interval (I) if f(x1) > f(x2) whenvever x1 < x2

4. What is a function ?

4.1. - A Function (f) is a rule that assigns to each element (x) in a set called (D) exactly one element, called f(X), in a set (E).

4.2. - Domain of a function = All possible inputs

4.3. - Range of a function = All possible outputs

5. Representation of Functions

5.1. - there are four ways to represent a function

5.1.1. 1- Verbally (by a description in words)

5.1.2. 2- Numerically (by a table of values)

5.1.3. 3- Visually (by a graph)

5.1.4. 4- Algebraically (by an explicit formula)

5.2. - The vertical line test : A curve in the xy-plane is the graph of a function of x if and only if no vertical line intersects the curve more than once.