Indicies - Explanation, Examples and Laws of Indicies

 

Relations and Functions.

In mathematics, relations and functions describe how one set of numbers connects to another. They form the foundation of algebra and are essential for understanding graphs, equations, and real-life mathematical models.

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What Is a Relation?

A relation is a rule or connection that pairs elements from one set (called the domain) with elements from another set (called the range).

If we have two sets:

• Set A = {1, 2, 3}

• Set B = {4, 5, 6}

A relation from A to B can be written as a set of ordered pairs such as:

$$R = \{(1, 4), (2, 5), (3, 6)\}$$

This means:

• 1 is related to 4

• 2 is related to 5

• 3 is related to 6

Each pair shows how an element from the first set connects to one from the second.

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Ways of Representing a Relation

Relations can be represented in several forms:

1. Set of Ordered Pairs: e.g., $R=\{(2,3),(3,4),(4,5)\}$

2. Mapping Diagram: uses arrows to show how each element in the domain is connected to elements in the range.

3. Graph: plots the ordered pairs on a coordinate plane.

4. Equation: for example, $y = 2x + 1$ defines a relation between $x$ and $y$.

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Domain and Range

• The domain is the set of all possible input values (x-values).

• The range is the set of all possible output values (y-values).

Example:
If $R=\{(1,2),(2,3),(3,4)\},<span\; style="font-size:\; medium;"><br></span>$ then the domain = {1, 2, 3} and the range = {2, 3, 4}.

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Types of Relations

1. One-to-One Relation:
   Each element in the domain maps to one unique element in the range.
   Example: $R=\{(1,2),(2,3),(3,4)\}$

2. One-to-Many Relation:
   One element in the domain is related to more than one element in the range.
   Example: $R=\{(2,3),(2,4)\}$

3. Many-to-One Relation:
   Two or more elements in the domain are related to the same element in the range.
   Example: $R = \{(1, 5), (2, 5), (3, 5)\}$

4. Many-to-Many Relation:
   Multiple elements in the domain relate to multiple elements in the range.

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What Is a Function?

A function is a special type of relation where each element of the domain is related to exactly one element in the range. This means no two ordered pairs in a function can have the same first element.

In simpler terms:
A function assigns one and only one output to each input.

Example:

$$f = \{(1, 3), (2, 4), (3, 5)\}$$

is a function because each input (1, 2, 3) has only one output.

However,

$$g=\{(1,3),(1,4),(2,5)\}$$

is not a function because input 1 maps to two different outputs (3 and 4).

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Function Notation

Functions are often written using letters such as $f, g, h$. If $y$ depends on $x$, we write it as:

$$y=f(x)$$

This means:
“$y$ is the value of the function $f$ when $x$ is the input.”

Example:
If $f(x) = 2x + 3$, then:

• $f(1) = 2(1) + 3 = 5$
  

• $f(2) = 2(2) + 3 = 7$
  

So the relation between $x$ and $y$ is functional - each $x$ gives exactly one $y$.

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Identifying a Function from a Relation

A relation is a function if no two ordered pairs have the same first element.

Examples:

1. $R_1 = \{(2, 5), (3, 6), (4, 7)\}$→ Function 

2. $R_2 = \{(1, 2), (1, 3), (2, 4)\}$ → Not a function  (1 maps to both 2 and 3)

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Vertical Line Test (Graph Method)

When relations are graphed on a coordinate plane:

• If a vertical line crosses the graph only once at any point, the relation is a function.

• If it crosses more than once, it is not a function.

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Types of Functions

1. Linear Function:
   $f(x) = mx + c$
   Example: $f(x) = 2x + 3$
   

2. Quadratic Function:
   $f(x) = ax^2 + bx + c$
   Example: $f(x) = x^2 + 2x + 1$
   

3. Cubic Function:
   $f(x) = ax^3 + bx^2 + cx + d$
   

4. Constant Function:
   $f(x)=\mathrm{k<span\; style="font-size:\; medium;"> </span>}$$k$ k is a constant.

5. Reciprocal Function:
   $f(x) = \frac{1}{x}$

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Worked Examples

Example 1:
Which of the following relations are functions?
a) $R = \{(1, 2), (2, 3), (3, 4)\}$
b) $R = \{(2, 4), (2, 5), (3, 6)\}$

Solution:
a) Each input has one output → Function 
b) Input 2 has two outputs (4 and 5) → Not a Function 

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Example 2:
If $f(x) = 3x - 2$, find $f(0)$, $f(2)$, and $f(-1)$.

Solution:

$$f(0) = 3(0) - 2 = -2$$
$$f(2) = 3(2) - 2 = 6 - 2 = 4$$
$$f(-1) = 3(-1) - 2 = -3 - 2 = -5$$

Hence,
$f(0) = -2, f(2) = 4, f(-1) = -5.$

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Example 3:
If $f(x) = x^2 + 1$, find the values of $f(2)$ and $f(-2)$.

Solution:

$$f(2) = (2)^2 + 1 = 4 + 1 = 5$$
$$f(-2)=(-2{)}^2+1=4+1=5$$

So, $f(2) = 5$ and $f(-2) = 5.$

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Assignments

1. Determine whether the following relations are functions:
   a) $\{(1, 4), (2, 5), (3, 6)\}$
   b) $\{(2, 3), (2, 4), (3, 5)\}$
   c) $\{(0, 1), (1, 1), (2, 1)\}$
   

2. Given $f(x) = 2x + 5$, find:
   a) $f(0)$

   
   

   b) $f(3)$

   
   

   c) f(−2)

3. $f(x) = x^2 - 4x + 3$, calculate:
   a) $f(1)$
   b) $f(2)$ 

4. Write the domain and range of: f(x) =x+1  for x={1,2,3,4}