Function comes in many shapes and sizes. Different types of functions have different properties that make them special. Suppose you are given a function but you don't know which type of function it is. Therefore, you will be working blindfolded. That is why it becomes necessary to understand which type of function you are working on.

To make things easy, we made a small hierarchy of functions.

 

In other words, functions are divided into two parts, algebraic and transcendental functions. Both functions are also divided into subgroups. In the case of algebraic, there are two types, one is the polynomial, and the other is the piecewise function. Furthermore, you can divide polynomial into different types. One is the constant type, the other is the linear, and last but not least, it is quadratic.

Since the algebraic part is done, let's learn about transcendental functions. The transcendental function can be divided into three types which are exponential, logarithmic, and trigonometric. If you want to learn more about them, keep on reading.

Explicit Functions

Functions that relate the independent variable with the dependent variable. The images of x are obtained by simple substitution.

y = 5x - 2

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Implicit Functions

It is the opposite of explicit function. This type of function contains both, independent as well as dependent variables. The images of x cannot be obtained by simple substitution.

5x - y - 2 =0

Algebraic Functions

In algebraic functions, the operations to be made with the independent variable are addition, subtraction, multiplication, division, exponents, and roots.

Polynomial Functions

Polynomial functions are the functions that are defined by a polynomial.

y = { a }_{ 0 } + { a }_{ 1 } x + { a }_{ 2 } { x }^{ 2 } + { a }_{ 3 } { x }^{ 3 } + ... { a }^{ n } { x }^{ n }

Its domain is \mathbb{R}.

Constant Functions

The criterion is given by a real number.

y = b

The graph is a horizontal line parallel to the x-axis.

Linear Function

y = mx + c

Its graph is an oblique straight line, which is defined by two points of the function.

Quadratic Functions

y = a { x }^{ 2 } + bx + c

Its graph is a parabola.

Piecewise Functions

Piecewise functions are functions defined by different criteria, according to the intervals being considered.

Absolute value functions.

Floor function.

Ceiling function.

Sign function.

Rational Functions

The criterion is given by a quotient between polynomials:

y = \frac { { a }_{ 0 } + { a }_{ 1 } x + { a }_{ 2 } { x }^{ 2 } + { a }_{ 3 } { x }^{ 3 } + ... { a }^{ n } { x }^{ n } }{ { b }_{ 0 } + { b }_{ 1 } x + { b }_{ 2 } { x }^{ 2 } + { a }_{ 3 } { x }^{ 3 } + ... { b }^{ m } { x }^{ m } }

The domain is equal to \mathbb{R}, but does not include the values of x that would annul the denominator.

Radical Functions

The criterion is given by the variable x under the radical sign.

The domain of a radical function of the odd index is \mathbb{R}.

The domain of a radical function of even index is { a }^{ x } equal to all values that make the radical greater than or equal to zero.

Transcendental Functions

The independent variable appears as an exponent, an index of root, logarithmic or trigonometric ratios.

Exponential Functions

Given a positive real number, a, the function that every real number x which corresponds to the power is called an exponential function.

y = { a }^{ x }

Logarithmic Functions

The logarithmic function in base a is the inverse function of the exponential function in base a.

y = \log_{ a } x

a > 0, \qquad a \neq 1

Trigonometric Functions

Sine Function

y = \sin { x }

Cosine Function

y = \cos { x }

Tangent Function

y = \tan { x }

Cosecant Function

y = \csc { x }

Secant Function

y = \sec { x }

Cotangent Function

y = \cot { x }

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.