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Introduction, Functions | Algebra - Mathematics PDF Download

Have you ever thought that a particular person has particular jobs or functions to do? Consider the functions or roles of postmen. They deliver letters, postcards, telegrams and invites etc. What do firemen do? They are responsible for responding to fire accidents. In mathematics also, we can define functions. They are responsible for assigning every single object of one set to that of another.

Functions
A function is a relation that maps each element x of a set A with one and only one element y of another set B. In other words, it is a relation between a set of inputs and a set of outputs in which each input is related with a unique output. A function is a rule that relates an input to exactly one output.

Introduction, Functions | Algebra - Mathematics

It is a special type of relation. A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B and no two distinct elements of B have the same mapped first element.  A and B are the non-empty sets. The whole set A is the domain and the whole set B is codomain.

Representation
A function f: X →Y is represented as f(x) = y, where, (x, y) ∈ f and x ∈ X and y ∈ Y.
For any function f, the notation f(x) is read as “f of x” and represents the value of y when x is replaced by the number or expression inside the parenthesis. The element y is the image of x under f and x is the pre-image of y under f.

Introduction, Functions | Algebra - Mathematics

Every element of the set has an image which is unique and distinct. If we notice around, we can find many examples of functions.
If we lift our hand upward, it is a function. Waving our hand freely, it is a function.  A walk in a circular track, yes it is a type of function. Now you can think of other examples too! A graph can represent a function. The graph is the set of all pairs of the Cartesian product.
Does this mean that every curve in the world defines a function? No, not every curve drawn is a function. How to find it? Vertical line test. If any curve intercepts a vertical line at more than one point, it is a curve only not a function.

Solved Example for You

Problem: Which of the following is a function?

1. Introduction, Functions | Algebra - Mathematics

2. Introduction, Functions | Algebra - Mathematics

3. Introduction, Functions | Algebra - Mathematics

Solution: Figure 3 is an example of function since every element of A is mapped to a unique element of B and no two distinct elements of B have the same pre-image in A.

The document Introduction, Functions | Algebra - Mathematics is a part of the Mathematics Course Algebra.
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FAQs on Introduction, Functions - Algebra - Mathematics

1. What are functions in mathematics?
Ans. Functions in mathematics are a fundamental concept that relates each element from one set, called the domain, to a unique element in another set, called the range. They can be thought of as a process or rule that takes an input and produces a corresponding output.
2. How do functions work in mathematics?
Ans. Functions work by defining a relationship between the input values and the output values. The input values are plugged into the function, undergo a specific operation or transformation, and then produce the corresponding output values. This relationship is usually expressed through an equation or a graph.
3. Can a function have more than one output?
Ans. No, by definition, a function can only have one output for each input. Each input value in the domain must correspond to a unique output value in the range. If a single input value produces multiple output values, then it does not meet the criteria of being a function.
4. What is the difference between a function and an equation?
Ans. A function and an equation are related but distinct concepts in mathematics. A function represents a relationship between the input and output values, while an equation represents a statement of equality between two expressions. Functions can be described by equations, but not all equations represent functions. In a function, each input has a unique output, while an equation can have multiple solutions or variables.
5. How are functions useful in real-life applications?
Ans. Functions play a crucial role in various real-life applications, such as physics, engineering, economics, and computer science. They help model and analyze relationships between different quantities, such as distance and time, cost and revenue, or temperature and pressure. By understanding and utilizing functions, we can make predictions, solve problems, and make informed decisions based on mathematical relationships.
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