Types of Functions

Types of Functions

Explicit Functions

If the images of x are obtained by simple substitution.

y = 5x − 2

Implicit Functions

If 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 = a0 + a1x + a2x² + a3x³ +··· + anxn

Its domain is Any Real Number.

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 + b

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

Quadratic Functions

y = ax² + 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:

Rational Functions

The domain is equal to 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 odd index is Any Real Number.

The domain of a radical function of even index is 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 ax is called an exponential function.

Exponential Function

Logarithmic Functions

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

Logarithmic Functions

Logarithmic Function Criteria

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