Functions

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Functions by Mind Map: Functions

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

5.5. (Special case)