# Types of Functions based on Domain

**Types of Functions based on Domain**

Functions are practiced in many other topics of maths. The function equations usually hold algebraic representations, trigonometric, logarithms and exponents and therefore are named based on these domain values. Check out more about the domain-based functions in this section.

**Trigonometric Functions**

The six basic trigonometric functions are sinθ, cosθ, tanθ,cotθ, secθ, cosecθ. Here abouts the domain value θ is the angle and is measured in degrees or radians. These trigonometric functions can be defined through the ratio of the sides of a right-angle triangle, via the Pythagoras theorem.

The trigonometric functions along with the inverse trigonometric functions are also sometimes included in periodic functions as the principal values are repeated after a fixed period of time.

**Inverse Trigonometric Functions:**

Trigonometric functions are many-one functions but we know that inverse of function exists if the function is bijective. If we restrict the domain of trigonometric functions, then these functions become bijective and the inverse of trigonometric functions are defined within the restricted domain. The inverse of f is denoted by f ^{-1}.

Let y = f(x) = sin x, then its inverse is x = sin^{-1} y.

**Domain and Range of Inverse Trigonometric Functions**

sin^{-1}(sinθ) = θ; ∀ θ ∈ [−*π**/*2,*π/*2]

cos^{-1}(cosθ) = θ; ∀ θ ∈ [0, π]

tan^{-1}(tanθ) = θ; ∀ θ [−*π**/*2,*π/*2]

cosec^{-1}(cosecθ) = 0; ∀ θ ∈ [−*π**/*2,*π/*2] , θ ≠ 0

sec^{-1}(secθ) = θ; ∀ θ ∈ [0, π], θ ≠ *π/*2

cot^{-1}(cotθ) = θ; ∀ θ ∈ (0, π)

sin(sin^{-1} x) = x, ∀ x ∈ [-1, 1]

cos(cos^{-1} x) = x; ∀ x ∈ [-1, 1]

tan(tan^{-1}x) = x, ∀ x ∈ R

cosec(cosec^{-1}x) = x, ∀ x ∈ (-∞, -1] ∪ [1, ∞)

sec(sec^{-1} x) = x, ∀ x ∈ (-∞, -1] ∪ [1, ∞)

cot(cot^{-1} x) = x, ∀ x ∈ R

Note: sin^{-1}(sinθ) = θ ; sin^{-1} x should not be confused with (sinx)^{-1} = 1/*sinx* or sin^{-1} x = sin^{-1}(1/x) and similarly for other trigonometric functions.

The value of an inverse trigonometric function, which lies in the range of principal value branch, is called the principal value of the inverse trigonometric function.

Note: Whenever no branch of an inverse trigonometric function is mentioned, it means we have to consider the principal value branch of that function.

**Properties of Inverse Trigonometric Functions**

**Algebraic Function**

A function that involves a finite number of terms including powers, roots of independent variable x, coefficient, constant term, plus fundamental operations like addition, multiplication, subtraction, and division are recognized as an algebraic equation. f(x)=6x^{3}-2x^{2}+4x+9

An algebraic function is essential to determine the various operations of algebra and is also identified as a linear function, cubic function, quadratic function, or polynomial function, depending on the degree of the algebraic equation.