A function in mathematics is a relationship between two sets, where each element in the first set (called the domain) is paired with exactly one element in the second set (called the codomain).
Functions are denoted by symbols like f(x), where x is the input from the domain, and f(x) is the output in the codomain.
Definition of Function
A function is like a machine that takes an input, does something to it, and produces a unique output. In More Detailed – A function is a relationship between two sets, called the domain and the codomain (or range), where each element in the domain is associated with exactly one element in the codomain.
Key Concepts of Function
- Domain: The set of all possible inputs for the function.
- Codomain (Range): The set of all possible outputs of the function.
- Unique Output: For every input, there can be only one output. This is the most important condition for a relationship to be considered a function.
Ways to Represent Functions
- Equations: y = f(x) (e.g., y = 2x + 3)
- Graphs: Visual representation on a coordinate plane.
- Tables: Listing inputs and their corresponding outputs.
- Mappings: Using arrows to show the association between inputs and outputs.
Example of Function
Let’s say we have a function f(x) = x + 2
- If we input x = 3, the function gives us an output of f(3) = 3 + 2 = 5.
- If we input x = -1, the function gives us an output of f(-1) = -1 + 2 = 1.
No matter what input we give this function, we’ll always get a single, unique output.
Types of Functions
Based on Mapping/Elements –
One-to-One (Injective) Function:
Each element in the domain maps to a unique element in the codomain. No two inputs have the same output.
- Example: f(x) = x + 5
Many-to-One Function:
Two or more elements in the domain map to the same element in the codomain.
- Example: f(x) = x² (both 2 and -2 map to 4)
Onto (Surjective) Function:
Every element in the codomain is mapped to by at least one element in the domain. The range is equal to the codomain.
- Example: If the codomain is all real numbers, f(x) = 2x is onto.
One-to-One and Onto (Bijective) Function:
The function is both one-to-one and onto. There’s a perfect pairing between inputs and outputs.
- Example: f(x) = x
Into Function:
If the function is not onto. There are elements in the codomain that are not mapped to by any element in the domain.
Constant Function:
Every input maps to the same output.
- Example: f(x) = 7
FAQs on Function :
What is the difference between a relation and a function?
The key difference between a relation and a function lies in how they associate inputs and outputs:
Relation:
- A relation is simply a set of ordered pairs. It describes any kind of association between elements of two sets.
- It can have one-to-one, one-to-many, many-to-one, or many-to-many mappings between inputs and outputs.
- Example: {(1, a), (1, b), (2, c)} – Here, the input ‘1’ is associated with two different outputs, ‘a’ and ‘b’.
Function:
- A function is a special type of relation with a crucial restriction: each input (x-value) must have exactly one output (y-value).
- It can have one-to-one or many-to-one mappings, but not one-to-many.
- Example: {(1, a), (2, b), (3, c)} – Each input has a unique output.
In simpler terms:
Imagine a vending machine.
- Relation: You press a button (input), and sometimes you get two different items (outputs) at once. That’s a relation, but not a very reliable vending machine!
- Function: You press a button (input), and you always get one specific item (output). That’s a function.
What is the domain and range of a function?
- Domain: The set of all possible inputs (x) for the function.
- Range: The set of all possible outputs (f(x)).
How do you determine if a graph represents a function?
Use the vertical line test: If any vertical line intersects the graph at more than one point, it is not a function.
Can a function have multiple inputs for the same output?
Yes, a function can map different inputs to the same output.
What is an inverse function?
An inverse function “undoes” the action of another function.
