Functions Formulae AA HL only

At its core, a mathematical function is a rule or a mapping that associates each element of a set (called the domain) with exactly one element of another set (called the codomain or range). It defines a dependent relationship where the output value is uniquely determined by the input value. We often write it as \(f: A \to B\), meaning function \(f\) maps elements from set \(A\) to set \(B\).

Formally, a function \(f\) from a set \(X\) to a set \(Y\) is a relation such that for every element \(x\) in \(X\), there is exactly one element \(y\) in \(Y\) with the property that \((x, y)\) is in the relation. We write \(y = f(x)\).

In more advanced mathematics, particularly in fields like linear algebra, functional analysis, and calculus of variations, the term "functional" has a specific meaning that differs from a standard "function". A functional is a map from a vector space (often a space of functions) to its underlying field of scalars (usually real or complex numbers). In simpler terms, a functional takes a function as its input and outputs a single number.

Examples:

• Integration: The definite integral \(\int_a^b f(x) dx\) is a functional, as it takes a function \(f\) and outputs a number.
• Evaluating a function at a point: For a fixed \(x_0\), the map \(f \mapsto f(x_0)\) is a functional.

While every functional is a type of function (mapping one set to another), the term "functional" is usually reserved for these cases where the input is itself a function or element from a function space.

A relation is a set of ordered pairs. It simply shows a connection or relationship between elements of two sets. For example, the relation "is less than" on the set {1, 2, 3} could be represented by the pairs {(1, 2), (1, 3), (2, 3)}.

A function is a specific type of relation. The key difference is the "exactly one output" rule. In a function, for any given first element in an ordered pair (the input), there is only one possible second element (the output). In the "is less than" example above, the input 1 is related to *both* 2 and 3, so it's a relation, but not a function from {1, 2, 3} to {1, 2, 3} in the standard sense.

Think of it this way: All functions are relations, but not all relations are functions.

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

• Solving \(f(x) = c\): This means finding the input value(s) \(x\) that produce a specific output \(c\). If \(f(x) = 2x+1\) and you want to solve \(f(x)=7\), you solve \(2x+1=7\), which gives \(x=3\).
• Finding Roots/Zeros (\(f(x) = 0\)): This is a common type of solving where you find the input values \(x\) that make the function's output equal to zero. These are the points where the function's graph crosses the x-axis.
• Solving Equations with Functions: This involves finding values (often of a variable) that satisfy an equation where functions are involved, e.g., solving \(f(x) = g(x)\) means finding where the graphs of \(f\) and \(g\) intersect.

The methods used depend heavily on the type of function (linear, quadratic, exponential, etc.).

A linear function is a polynomial function of degree zero or one. Its graph is always a straight line. The most common form is \(f(x) = mx + b\), where \(m\) is the slope of the line (how steep it is) and \(b\) is the y-intercept (where the line crosses the y-axis). A horizontal line (\(m=0\), \(f(x) = b\)) is also a linear function (a constant function).

'r' can represent various things depending on the context, but it's commonly used:

• As a variable, just like 'x' or 'y'.
• In polar coordinates, 'r' represents the distance from the origin.
• In formulas involving circles or spheres, 'r' often represents the radius.
• In statistics, 'r' can represent the correlation coefficient.

Without a specific function or context, 'r' is usually just another variable representing an unknown or changing number.

Functional analysis is a branch of mathematics that originated from the study of spaces of functions. It focuses on vector spaces endowed with some kind of limit-related structure (like a norm, inner product, or topology) and the linear operators acting upon these spaces. It's a fundamental area in advanced mathematics with applications in differential equations, quantum mechanics, and signal processing. Functionals (as described above) are key objects studied in functional analysis.

The Gamma function (\(\Gamma(z)\)) is an extension of the factorial function to complex numbers. For positive integers \(n\), \(\Gamma(n) = (n-1)!\). For complex numbers \(z\) with a positive real part, it's defined by the integral \(\Gamma(z) = \int_0^\infty t^{z-1}e^{-t} dt\). It appears in various areas of mathematics and physics.

Hyperbolic functions are a set of functions related to the hyperbola, analogous to how trigonometric functions are related to the circle. The basic hyperbolic functions are the hyperbolic sine (sinh), hyperbolic cosine (cosh), and hyperbolic tangent (tanh). They are defined using the exponential function:

• \( \text{sinh}(x) = \frac{e^x - e^{-x}}{2} \)
• \( \text{cosh}(x) = \frac{e^x + e^{-x}}{2} \)
• \( \text{tanh}(x) = \frac{\text{sinh}(x)}{\text{cosh}(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}} \)

They appear in calculus, differential equations, and various engineering and physics problems.