Functions Formulae AI HL only

A mathematical function is a rule that establishes a relationship between two sets, typically called the domain (the set of input values) and the codomain (the set containing the output values). The crucial characteristic of a function is that it assigns exactly one output value from the codomain to each input value from the domain.

This one-to-one assignment (for inputs) is what distinguishes a function from a more general mathematical relation.

A relation is any set of ordered pairs connecting elements from one set to elements of another. For example, the relation "is the child of" between a set of children and a set of parents.

A function is a specific type of relation where each element in the first set (the domain) is paired with exactly one element in the second set. If the relation "is the mother of" is considered from children to mothers, it's a function because each child has only one biological mother. However, "is the parent of" from children to parents is a relation but not a function (a child can have two parents).

All functions are relations, but not all relations are functions.

Functions can be categorized based on various properties or the form of their rule:

• Based on Rule Structure: Algebraic (linear, quadratic, polynomial, rational, etc.), Transcendental (exponential, logarithmic, trigonometric).
• Based on Mapping Property: One-to-One (Injective), Onto (Surjective), Bijective (Both one-to-one and onto).
• Based on Behavior: Continuous, Discrete, Even, Odd, Periodic.
• Based on Combinations: Composite functions, Inverse functions.

In Discrete Mathematics, functions are defined using the same core concept: a mapping from a domain set to a codomain set where each element in the domain maps to exactly one element in the codomain. The difference is that the sets involved (domain and codomain) are often discrete sets, meaning they consist of separate, distinct elements (like integers, finite sets, or sets of objects). Discrete math functions are fundamental to topics like set theory, graph theory, and computer science.

A linear function is a type of algebraic function that produces a straight line when graphed. It has the general form \(f(x) = mx + b\), where \(m\) is the slope (rate of change) and \(b\) is the y-intercept (value of \(f(x)\) when \(x=0\)).

A bijective function (or bijection) is a function that is both:

• One-to-One (Injective): Every distinct element in the domain maps to a unique element in the codomain (no two different inputs map to the same output).
• Onto (Surjective): Every element in the codomain is mapped to by at least one element in the domain (the range of the function is equal to its codomain).

Bijective functions are important because they have a well-defined inverse function.

In discrete mathematics and computer science, a Boolean function is a function whose domain is a set of binary values (usually \(\{0, 1\}\) or \(\{\text{True, False}\}\)) and whose codomain is also a set of binary values. These functions are fundamental in logic gates, digital circuits, and propositional logic.

Analytic functions are functions that can be locally represented as a convergent power series. In real analysis, a real analytic function is a function \(f\) defined on a subset of the real numbers such that for any \(x_0\) in the domain, \(f(x)\) can be represented by a power series that converges to \(f(x)\) in a neighborhood of \(x_0\). In complex analysis, complex analytic functions (also called holomorphic functions) are functions defined on a subset of the complex numbers that are complex differentiable at every point. Analytic functions are very well-behaved and have strong properties.

You typically don't "solve" a function itself. You solve equations involving functions, find specific values (evaluate), or analyze their properties.

• Evaluate \(f(a)\): Substitute a value \(a\) into the function rule to find the output \(f(a)\).
• Solve \(f(x) = c\): Find the input value(s) \(x\) that result in a specific output \(c\). This is solving an equation.
• Find Roots/Zeros (\(f(x) = 0\)): A common type of solving where you find the inputs \(x\) that make the function's output zero.

The methods used depend on the function type (algebraic techniques for polynomials, graphical methods, numerical methods, calculus for optimization, etc.).

Something is not a function if, for at least one input value in its domain, there is more than one output value. Graphically, this means it fails the "vertical line test" - if you can draw a vertical line that crosses the graph at more than one point, it's not a function.

Examples of things that are relations but not functions:

• The relationship \(x = y^2\). For an input \(x=4\), the output \(y\) could be 2 or -2.
• The graph of a circle.
• A list of pairs like \(\{(1, 5), (2, 8), (1, 10)\}\) - input 1 has two outputs.