1. Definition
1.1. Domain: Input/ X axis Range: Output/ Y axis
1.2. f: X → Y
1.2.1. Relation which maps each element x in the set of X to one and only one element y in the set of Y
2. Functions
2.1. Existence
2.1.1. Vertical Line Test
2.1.1.1. A relation f is a function if and only if any vertical line, x=k∈Df, cuts the graph of f at one and only one point
2.1.1.2. Draw graph & Write explanation
2.1.1.3. Affirmative
2.1.1.3.1. "Since any vertical line x=k, k∈R (domain), cuts the graph of f at one and only one point, f is a function"
2.1.1.4. Negative
2.1.1.4.1. "Since the verticle line x=2, where 2∈R (domain), does not cut the graph of g at one and only one point, g is not a function"
2.1.2. Representation
3. One-One Functions
3.1. A function f is said to be one-one if every element in the domain of f has an image and no two elements have the same image
3.2. Existence
3.2.1. Horizontal Line Test
3.2.1.1. Affirmative
3.2.1.1.1. "Since any horizontal line y=k, k≠-1 cuts the graph of f at one and only one point, therefore f is a one-one function"
3.2.1.2. Negative
3.2.1.2.1. "Range of g = (0, 2]. Since the horizontal line y=1 does not cut the graph of g at one and only one point, g is not a one-one function."
3.2.1.3. Sketch graph & Write explanation
3.3. Domain restriction to obtain a one-one function (through graph)
4. Inverse Funtions
4.1. Exists if is one-one
4.2. DOMAIN, and RANGE
4.3. Obtaining RULE: Make x the subject of the formula
4.4. Graphical relationship between a one-one function and its inverse: Reflections about line y=x
4.5. Domain restriction to obtain an inverse function
5. Composite Functions
5.1. Definition
5.2. The composite function gf exists, if the range of f is a subset of the domain of g
5.3. Domain
5.4. Range
5.4.1. Sketch graph of gf and read off its range
5.4.2. Sketch graphs of g and f