Functions Formulae AI SL & AI HL

A mathematical function is a rule or a relationship that pairs each element from one set (called the domain) with exactly one element from another set (called the codomain, or more specifically, the range of the function). Think of it as a well-behaved machine: for every valid input you give it, it produces one unique output.

We often write a function as \(f(x)\), where \(x\) is the input variable, \(f\) is the function rule, and \(f(x)\) is the output value. The relationship must satisfy two conditions:

• Every input in the domain must be associated with an output.
• Each input must be associated with *only one* output.

A relation is simply a set of ordered pairs that shows a connection between elements of two sets. For example, \(\{(1, 2), (1, 3), (2, 4)\}\) is a relation.

A function is a special type of relation where each element in the first set (the domain) is paired with *exactly one* element in the second set (the codomain). Using the example above, \(\{(1, 2), (1, 3), (2, 4)\}\) is a relation but not a function because the input 1 is paired with two different outputs, 2 and 3.

Every function is a relation, but not every relation is a function.

Functions can be classified in many ways based on their properties, rules, or graphs. Some common types include:

• Algebraic Functions: Formed using algebraic operations (addition, subtraction, multiplication, division, roots, powers). Examples: Linear, Quadratic, Polynomial, Rational.
• Transcendental Functions: Functions that are not algebraic. Examples: Exponential, Logarithmic, Trigonometric.
• Specific types based on behavior: Continuous, Discrete, Even, Odd, One-to-One (Injective), Onto (Surjective), Bijective, Inverse, Composite.

A linear function is an algebraic function where the highest power of the variable is 1. Its graph is a straight line. It can be written in the form \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Constant functions (like \(f(x) = 5\)) are also considered linear functions.

A simple example of a function is \(f(x) = 2x + 1\). Here, \(x\) is the input, and the rule is "multiply the input by 2, then add 1".

• If the input is \(x = 3\), the output is \(f(3) = 2(3) + 1 = 7\).
• If the input is \(x = -5\), the output is \(f(-5) = 2(-5) + 1 = -10 + 1 = -9\).

Every input \(x\) gives exactly one output \(f(x)\). The domain could be all real numbers, and the range would also be all real numbers for this specific function.

In more advanced mathematics (like functional analysis), a "functional" is a special type of mapping that takes a function (or an element from a function space or vector space) as its input and returns a single scalar value (usually a real or complex number) as its output. While technically a function, the term "functional" distinguishes it from functions whose inputs are numbers.

Example: The definite integral \(\int_a^b f(x) dx\) is a functional, as it takes a function \(f\) and gives a number as a result.

Functional analysis is a branch of mathematics that studies vector spaces of functions and related concepts like linear operators and functionals on these spaces. It provides a framework for studying infinite-dimensional spaces and has applications in areas like quantum mechanics, differential equations, and signal processing. Functionals are central objects of study in this field.

The phrase "solving functions" isn't standard terminology. You typically "solve equations involving functions," "evaluate functions," or "find the roots (or zeros) of a function."

• Evaluate \(f(a)\): Find the output when the input is \(a\). Substitute \(a\) for the variable in the function's rule.
• Solve \(f(x) = c\): Find the input value(s) \(x\) that produce a specific output \(c\). This involves setting the function's expression equal to \(c\) and solving the resulting equation for \(x\).
• Find Roots/Zeros (\(f(x) = 0\)): Find the input value(s) \(x\) where the function's output is 0. This means solving the equation \(f(x) = 0\).

The specific methods for solving depend on the type of function (linear, quadratic, etc.).

How you define or write a mathematical function depends on the computational tool or programming language you are using:

• Mathematica: Use a pattern with an underscore. Example: f[x_] := 2x + 1
• Python: Use the def keyword. Example:

  def f(x):
      return 2*x + 1

• MATLAB: Use an anonymous function or define a function in a script/file. Example (Anonymous Function): f = @(x) 2*x + 1;

In general mathematical writing, you state the function rule, e.g., "Let \(f(x) = 2x + 1\) for all real numbers \(x\)."