Prab: CSV import

2025-03-17T13:11:44Z

CSV import

← Older revision		Revision as of 13:11, 17 March 2025
Line 90:		Line 90:
	[[Category:Mathematical Concepts]]		[[Category:Mathematical Concepts]]
	{{No image}}		{{No image}}
			__NOINDEX__

Prab: CSV import

2025-02-10T17:51:47Z

CSV import

← Older revision		Revision as of 17:51, 10 February 2025
Line 89:		Line 89:
	[[Category:Functions]]		[[Category:Functions]]
	[[Category:Mathematical Concepts]]		[[Category:Mathematical Concepts]]
			{{No image}}

Prab: CSV import

2024-10-27T18:22:23Z

CSV import

New page

= Functions in Mathematics =

Functions are fundamental concepts in mathematics, serving as the building blocks for understanding relationships between varying quantities. A function is a relation between a set of inputs and a set of permissible outputs, with the property that each input is related to exactly one output.

== Definition ==
A function \( f \) from a set \( X \) to a set \( Y \) is defined as a relation that assigns to each element \( x \) in \( X \) exactly one element \( y \) in \( Y \). This is often denoted as:

\[
f: X \to Y
\]

where \( f(x) = y \).

== Notation ==
Functions are commonly denoted by letters such as \( f \), \( g \), or \( h \). The notation \( f(x) \) represents the output of the function \( f \) corresponding to the input \( x \).

== Types of Functions ==

=== Linear Functions ===
A linear function is a function of the form:

\[
f(x) = mx + b
\]

where \( m \) and \( b \) are constants. The graph of a linear function is a straight line.

=== Quadratic Functions ===
A quadratic function is a function of the form:

\[
f(x) = ax^2 + bx + c
\]

where \( a \), \( b \), and \( c \) are constants. The graph of a quadratic function is a parabola.

=== Polynomial Functions ===
A polynomial function is a function that can be expressed in the form:

\[
f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0
\]

where \( a_n, a_{n-1}, \ldots, a_0 \) are constants and \( n \) is a non-negative integer.

=== Exponential Functions ===
An exponential function is a function of the form:

\[
f(x) = a^x
\]

where \( a \) is a positive constant.

=== Logarithmic Functions ===
A logarithmic function is the inverse of an exponential function and is of the form:

\[
f(x) = \log_a(x)
\]

where \( a \) is the base of the logarithm.

== Properties of Functions ==

=== Domain and Range ===
The domain of a function is the set of all possible inputs for the function, while the range is the set of all possible outputs.

=== Injective, Surjective, and Bijective Functions ===
A function is called:
* **Injective** (or one-to-one) if different inputs produce different outputs.
* **Surjective** (or onto) if every element in the output set is mapped to by at least one input.
* **Bijective** if it is both injective and surjective, meaning it establishes a one-to-one correspondence between the input and output sets.

== Applications ==
Functions are used in various fields such as physics, engineering, economics, and biology to model relationships between quantities. For example, in [[physics]], functions describe the motion of objects, while in [[economics]], they model supply and demand relationships.

== See Also ==
* [[Calculus]]
* [[Algebra]]
* [[Trigonometry]]
* [[Differential Equations]]

== References ==
* Stewart, James. "Calculus: Early Transcendentals." Cengage Learning.
* Larson, Ron. "Precalculus." Cengage Learning.

[[Category:Mathematics]]
[[Category:Functions]]
[[Category:Mathematical Concepts]]

Functions - Revision history

Prab: CSV import

Prab: CSV import

Prab: CSV import