Functions

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.