# Functions

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Functions by

## 1. Definition

### 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

### 3.2. 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

## 5. Composite Functions

### 5.4. Range

5.4.1. Sketch graph of gf and read off its range

5.4.2. Sketch graphs of g and f