Functions

A comprehensive set of vocabulary flashcards derived from the lecture notes on functions and trigonometric functions, covering key concepts and definitions.

Sets of Numbers

Categories of numbers that includes natural numbers, integers, rational numbers, and real numbers.

Natural Numbers (N)

The positive whole numbers: {1, 2, 3, 4, …}.

Integers (Z)

All whole numbers including both positive and negative numbers and zero: {…, -3, -2, -1, 0, 1, 2, 3, …}.

Rational Numbers (Q)

Numbers that can be expressed as fractions of two integers: e.g. 1/2, -5/4.

Real Numbers (R)

All rational numbers and other numbers that cannot be expressed as fractions, including √2 and π.

Element of a Set

A number that belongs to a set, denoted by the symbol ∈.

Subset

A set containing some, but not all, elements of another set. Denoted by $\subset$

Interval Notation

A method to represent subsets of real numbers, indicating ranges between two values.

Closed Interval [a, b]

Set of all x such that a ≤ x ≤ b.

Open Interval (a, b)

Set of all x such that a < x < b.

Union of Sets (∪)

Combines two sets to include all elements from both.

Function

A rule that assigns each element in a domain D a unique value in a set S.

Domain of a Function (D)

The set of all possible inputs for a function.

Range of a Function (R(f))

The set of all output values produced by the function.

Rational Function

A function expressed as the ratio of two polynomial functions, $p(x)/q(x)$, where q(x) ≠ 0.

Exponential Function

Functions of the form $f(x) = a^x$ where a is a positive real number.

Logarithmic Function

The inverse of an exponential function, defining the exponent required to achieve a value.

Trigonometric Functions

Functions such as sin, cos, and tan, relating angles to ratios of side lengths in right triangles.

Sinusoidal Wave

A mathematical curve that describes a smooth periodic oscillation.

Amplitude (A)

Half the distance between the maximum and minimum values of a wave.

Period (T)

The length of time it takes for one complete cycle of a wave.

Frequency (f)

The number of cycles that occur in a unit of time, measured in Hertz (Hz).

Phase Shift

The horizontal shift of the periodic wave, affecting its starting point.

Piecewise Function

A function defined by multiple sub-functions, each applying to a specific interval.

Heaviside Function

A piecewise function defined as H(x) = 0 for x < 0, 1 for x ≥ 0.

Absolute Value Function

A function defined as f(x) = |x| = x if x ≥ 0 and f(x) = -x if x < 0.

Hyperbolic Functions

Functions defined in terms of the exponential function, such as sinh, cosh, and tanh.

Combination of Functions

The result of applying operations (addition, subtraction, multiplication, and division) to functions.

Composite Functions (f ◦ g)

Combining two functions where the output of one function becomes the input of another.

Vertical Shift of a Function

A transformation that shifts the graph of a function up or down.

Constant Function

A function that always returns the same value, regardless of the input.

Contour Plot

A graphical representation showing level curves of a function in a two-dimensional plane.

Fourier Analysis

A mathematical method to decompose functions into a sum of sine and cosine functions.

Dynamic Systems

Models to describe systems that change over time based on their variables.

Sinusoidal Function

Any function that can be written in the form A sin(ωt + φ).

Graph of a Function

A visual representation of the relationship between input and output values of a function.

Angular Frequency (ω)

The rate of change of the phase of a sinusoidal waveform, measured in radians per second.

Absolute Mean of a Function

The average of the absolute values of the function over its domain.

Decay Rate in Functions

A measurement of how quickly a function approaches zero over time.

Stability Conditions of Functions

Criteria that determine the behavior of a function under perturbations.