Functions

A function is a rule that assigns exactly one output to each input from a given set. Functions are central to all areas of mathematics, from basic algebra to advanced calculus and applied sciences.

Reference: Functions (Definitions, properties and examples),

Polynomial Functions

A polynomial function is made up of terms consisting of variables raised to whole-number powers and multiplied by coefficients.

f(x) = axⁿ + bx^n-1 + ... + k

Example: f(x) = 2x³ - 4x + 7

Reference: Polynomials,

Trigonometric Functions

Trigonometric functions relate angles to ratios of triangle sides and periodic wave patterns.

sin(x) - Sine
cos(x) - Cosine
tan(x) - Tangent

These functions are periodic and widely used in physics and engineering.

Reference: Trigonometric Functions,

Inverse Trigonometric Functions

Inverse trig functions find the angle whose trigonometric ratio is a given value:

sin^-1(x) or arcsin(x)
cos^-1(x) or arccos(x)
tan^-1(x) or arctan(x)

They are useful in solving triangles and modeling angles in geometry.

Reference: Inverse Trigonometric Functions,

Logarithmic and Exponential Functions

Exponential functions have the form f(x) = a^x, where a > 0. They model growth and decay.

Logarithmic functions are the inverses of exponentials: logₐ(x) is the exponent to which a must be raised to get x.

log(x) - Common log (base 10)
ln(x) - Natural log (base e, where e is approximately 2.718)

Reference: Logarithmic and Exponential Functions,

Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions but relate to hyperbolas rather than circles:

sinh(x) = (e^x - e^-x)/2
cosh(x) = (e^x + e^-x)/2
tanh(x) = sinh(x)/cosh(x)

These functions are used in calculus, physics, and engineering, especially in wave equations and relativity.

Reference: Hyperbolic Functions,

Factorial Function

The factorial function, denoted by n!, is the product of all positive integers less than or equal to n.

Example: 5! = 5 × 4 × 3 × 2 × 1 = 120

0! is defined as 1.

Factorials are used in permutations, combinations, and series expansions.

Reference: Factorial Function,

Conclusion

Functions provide a powerful framework for describing relationships between variables. Whether analyzing trends, solving equations, or modeling natural phenomena, functions like polynomials, trig, exponential, and factorial functions are essential tools in mathematics and beyond.