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Latest revision as of 13:11, 17 March 2025
Functions in Mathematics[edit]
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[edit]
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[edit]
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[edit]
Linear Functions[edit]
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[edit]
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[edit]
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[edit]
An exponential function is a function of the form:
\[
f(x) = a^x
\]
where \( a \) is a positive constant.
Logarithmic Functions[edit]
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[edit]
Domain and Range[edit]
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[edit]
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[edit]
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[edit]
References[edit]
- Stewart, James. "Calculus: Early Transcendentals." Cengage Learning.
- Larson, Ron. "Precalculus." Cengage Learning.
