What is a Relation?

A relation in algebra is a set of ordered pairs, where each element in the domain is related to an element in the range based on the a certain rule or condition. Relations are foundational to understanding functions that is, relations help us understand how elements from two sets interact with each other, and they serve as the foundation for studying functions.

Example 1: Simple Ordered Pairs

Consider two sets:

• Set A = {1, 2, 3} (domain)
• Set B = {4, 5, 6} (range)

A relation could be:

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

This relation pairs each element in Set A with an element in Set B could be a set of ordered pairs as folloes: R={(1,4),(2,5),(3,6)}

Example 2: A Non-Function Relation

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

This is a relation, but not a function. Because, the element 1 from the domain is paired with both 4 and 5, which means one element from the domain is related to multiple elements in the range. This is a relation but not a function since a function requires that each element in the domain map to exactly one element in the range.

Graphical Representation of a Relation

A relation can be represented as a graph of points (𝑥 , 𝑦) in a coordinate plane. For instance, the ordered pairs (1,4),(2,5),(3,6) from the earlier example can be plotted as points.

Conclusion

A relation is a broader concept that simply shows how two sets are connected, while a function is a special type of relation with stricter rules. Studying relations provides a foundation for understanding functions, which are pivotal in algebra.

What is a Function?

A function is a special type of relation where each input from the domain is related to exactly one output in the range. Functions are essential in algebra because they describe consistent relationships. The key distinction between a function and a general relation is that a function assigns only one output to each input.

Definition of Function

A function 𝑓 from set 𝐴 to set 𝐵 is a relation that associates each element 𝑥 in set 𝐴 (the domain) with exactly one element 𝑦 in set 𝐵 (the range). This can be written as 𝑓: 𝐴 → 𝐵, meaning that the function 𝑓 takes inputs from 𝐴 and assigns them to values in 𝐵.

n simpler terms:

• A function is a rule or a process that assigns each input exactly one output.
• If an input 𝑥 from the domain corresponds to an output 𝑦, we write 𝑓(𝑥)= 𝑦, where 𝑓(𝑥) is the output.

Function Notation

• A function is usually denoted by a letter such as 𝑓, 𝑔, or ℎ.
• If 𝑓 is a function, we write 𝑓(𝑥) = 𝑦, meaning that 𝑓 maps 𝑥 to 𝑦.
• The domain is the set of all possible inputs 𝑥, and the range is the set of all possible outputs 𝑓(𝑥).

Example 1: Simple Linear Function

Let f(x) = 2x + 3. This is a linear function where each input is associated with exactly one output.

Here,

• The domain is all real numbers (𝑥 ∈ 𝑅 ).
• The function rule 𝑓(𝑥) = 2𝑥 + 3 maps each input 𝑥 to an output given by multiplying 𝑥 by 2 and then adding 3.

For example:

• If x = 1, then f(1) = 2(1) + 3 = 5
• If x = 2, then f(-2) = 2(-2) + 3 = -1

This is a function because for each input 𝑥, there is only one corresponding output.

Example 2: Function in Real Life (Temperature Conversion)

Consider the function that converts Celsius to Fahrenheit: $$ f(c) = \frac{9}{5} C + 32 $$

In this case:

• The input is the temperature 𝐶 in Celsius, and the output is the temperature 𝑓(𝐶) in Fahrenheit.
• If C = 0, then $$ f(0) = \frac{9}{5} (0)+ 32 = 32 \quad degrees \quad Fahrenheit $$
• If C = 100, then $$ f(100) = \frac{9}{5} (100) + 32 = 212 \quad degrees\quad Fahrenheit $$

This function has one unique Fahrenheit value for every Celsius input, so it satisfies the definition of a function.

Example 3: Non-Function Example (Failing the Function Test)

Let’s revisit the relation: $$ R = {(1,4),(1,5), (2,6)}, $$

In this relation, 1 is related to both 4 and 5. This is not a function because the input 1 does not have a unique output. A function requires each input to map to exactly one output.

Vertical Line Test

One way to visually determine whether a relation is a function is to use the vertical line test on its graph. If a vertical line drawn through any point on the graph intersects the graph at more than one point, the relation is not a function. If it only intersects once for any vertical line, then it is a function.

Example 4: Graphical Representation

Consider the graph of: $$ f(x) = x^2 $$

• For each 𝑥 value, there is only one corresponding 𝑓(𝑥) value (e.g., 𝑓(2)= 4 and 𝑓(−2)=4).
• A vertical line will only intersect the graph at one point for each 𝑥-value, confirming that this is a function.

Now consider the graph of a circle, say $$ x^2 + y^2 = 1 $$

• For some 𝑥-values (e.g., 𝑥 = 0), there are two 𝑦-values (𝑦 = 1 and 𝑦 = − 1).
• A vertical line drawn at 𝑥 = 0 intersects the graph at two points, meaning the relation fails the vertical line test, so it is not a function.

Domain and Range of Functions

• The domain of a function is the set of all possible input values for which the function is defined.
• The range is the set of all possible output values.

For example, the function 𝑓(𝑥) = 1/𝑥 has a domain of all real numbers except 𝑥 = 0 (since division by zero is undefined). Its range is all real numbers except 𝑦 = 0.

Conclusion:

A function is a specific type of relation where each input is associated with exactly one output. Functions are fundamental in algebra because they describe predictable, consistent relationships between variables. Whether through equations, tables, or graphs, understanding functions allows us to model and solve real-world problems, making them an essential concept in algebra.

Function Notation

Function notation is a way to express the relationship between the input and the output in a concise format.

• f(x) means the function \( f \) is evaluated at input \( x \).
• The notation \( f(x) = y \) indicates that \( y \) is the output when \( x \) is the input.
• For example, if \( f(x) = 2x + 3 \), then \( f(2) = 2(2) + 3 = 7 \).