1. One-One Functions
1.1. 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
1.2. Existence
1.2.1. Horizontal Line Test
1.2.1.1. Affirmative
1.2.1.1.1. "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"
1.2.1.2. Negative
1.2.1.2.1. "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."
1.2.1.3. Sketch graph & Write explanation
1.3. Domain restriction to obtain a one-one function (through graph)
2. Composite Functions
2.1. Definition
2.2. The composite function gf exists, if the range of f is a subset of the domain of g
2.3. Domain
2.4. Range
2.4.1. Sketch graph of gf and read off its range
2.4.2. Sketch graphs of g and f
2.5. (Special case)
3. Definition
3.1. Domain: Input/ X axis Range: Output/ Y axis
3.2. f: X → Y
3.2.1. Relation which maps each element x in the set of X to one and only one element y in the set of Y
4. Functions
4.1. Existence
4.1.1. Vertical Line Test
4.1.1.1. 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
4.1.1.2. Draw graph & Write explanation
4.1.1.3. Affirmative
4.1.1.3.1. "Since any vertical line x=k, k∈R (domain), cuts the graph of f at one and only one point, f is a function"
4.1.1.4. Negative
4.1.1.4.1. "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"
4.1.2. Representation