Chapter 3: Function

Introduction:

In mathematics, understanding the concept of relations is essential before diving into functions. Let's start by defining a relation and then explore how it connects to a function.

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, that is, the relation is a correspondence between two sets: domain set and range set. 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.

Real-life scenario

    Examples of input output

• Vending Machine: You select a number (input), and the machine gives a specific snack (output).
• Temperature Conversion: If today’s temperature is given in Celsius, we can use a formula to convert it to Fahrenheit.
• Cell Phone Plans: A company charges a base fee plus a fixed cost per minute.

Related context

Linear Function (Constant Rate of Change)
• Example: 𝑓 ( 𝑥 ) = 2 𝑥 + 3
• Graph: A straight line
• Context: Taxi fare charges a base fee of \$3 plus \$2 per mile.
  Equation: Total Fare = 2 × (miles) + 3

Quadratic Function (Parabolic Growth)
• Example: $$ 𝑓 ( 𝑥 ) = 𝑥^2 − 4 𝑥 + 3 $$
• Graph: A parabola
• Context: Models projectile motion (e.g., throwing a ball—height vs. time).

Exponential Function (Rapid Growth or Decay)
• Example: $$ 𝑓 ( 𝑥 ) = 2^ 𝑥 $$
• Graph: An increasing curve
• Context: Population growth, interest rates, or virus spread.

Piecewise Function (Different Rules for Different Intervals)
• Example: $$ 𝑓 ( 𝑥 ) = 𝑥 + 2 , 𝑥 ≤ 3 $$ $$
      = 2 𝑥 − 1 , 𝑥 > 3 $$
• Context: Electricity bills with different rates for different usage levels

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.

Tabular representation of a Relation

A relation can be presented in a tabular form, such as, the column representing the domain set with the elements: {1, 2, 3}, and the second column, the range set with elements: {4, 5, 6}.

Relation represented by mapping

The relation can be represented by mapping by drawing an arrow from an element in the domain set to the corresponding element in the range set.

Relation in algebraic expression

The relation also can be expressed by an algebraic expression using the formula, such as: $$ f(x) = 300 - 0.65 x^2 $$

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.

Function:

A function is a special type of relation between two nonempty sets: domain and range, 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.

In 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, also called the value of the function $ f $ at $ x $ or the image of $ x $.
• The function assures the certainty, that is, for a given input there is certainty of output in a function, which is not possible if the relation is not a function.
• The concept of “one output” of a function is an important predictable behavior in a real time situations, which ensures the fixed amount of output for a given input.

Function Notation

• A function is usually denoted by a letter such as 𝑓, 𝑔, ℎ, ... etc..
• 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).

Function represented by mapping

The function can be represented by mapping by drawing an arrow from an element in the domain set to the corresponding element in the range set.

Function as a mapping!

Tabular representation of a Function

A function can be presented in a tabular form, such as, the column representing the domain set with the elements: {1, 2, 3}, and the second column, the range set with elements: {4, 5, 6}.

Function as a Tabular form!

Graphical Representation of a function

A function 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.

Function as a Graphical representation!

Function in algebraic expression

The Function also can be expressed by an algebraic expression using the formula, such as: $$ f(x) = 300 - 0.65 x^2 $$

Algebraic representation of function!

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 the value of 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
• and so on ...

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.