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# What are mathematical functions?

henry anorue

A function is an expression, rule, or law determining the relationship between independent and dependent variables in mathematics. Parts are widely used in mathematics to formulate physical connections in the natural sciences.

A function from a Set A to a Set B is an assignment of an element of B to each part of A. Set A is called the domain of the function, and set B is called the codomain of the operation.

Diagram of a function

You can also represent a function graphically, as shown below:

Diagram of a function

### Types of functions

Function classification makes it easy to understand and learn different types of functions.

The result can be presented as a function for any mathematical expression with an input value.

Functions can be classified into two categories:

• Elementary functions
• Special functions

### Elementary functions

Elementary functions include basic polynomials, trigonometric, hyperbolic, and exponential functions.

Elementary functions can further be divided into algebraic functions and non-algebraic functions.

### Algebraic function

Algebraic functions contain only fractions or rational exponents and algebraic operations such as addition, subtraction, multiplication, and division.

• Constant function: This is a zero degree polynomial, graph - horizontal line. For example, the function y(x)= 9 is a constant function.
• Linear function: This contains first-order polynomials. The graph is a straight line. A linear function has the form: y = f(x) = c + bx. y is the dependent variable. x is the independent variable. c is the y-intercept. b is the slope or gradient of the line.
• Quadratic function: A quadratic function has the form: f(x) = a$x^2$ + bx + c. Here, a, b, and c are numbers not equal to zero. It’s a second-degree polynomial and the graph is a parabola.
• Cubic function: This is a polynomial function of the third degree. It has the form f(x) = a$x^3$ + b$x^2$ + cx + d. Here, a, b, c, and d are real numbers. a ≠ 0.

### Non-algebraic function

A non-algebraic function is an analytic function that cannot be represented by a finite sequence of algebraic addition, subtraction, multiplication, division, and exponentiation.

• Exponential function: An exponential function is a mathematical function of the form: f(x)= $a^x$ . Here, x is a variable, and a is a constant called the function’s base. This base must be greater than zero.

• Hyperbolic function: This is similar to a regular trigonometric function but is defined using a hyperbola rather than a circle. The two main hyperbolic functions are sinh, sinh(x) = $\frac{e-e^{-x}}2$) and cosh, cosh(x) = $\frac{e + e^{-x}}2$).

• Logarithmic function: A logarithm is the inverse of the exponent. This means that the logarithm of a given number x is the exponent that another fixed number, the base b, must be raised to get x. The logarithmic function y = $log_b$x is equivalent to the exponential equation x = $b^y$.

• Periodic function: This is a trigonometric function. Examples include sine, cosine, tangent, cotangent, secant, and cosecant. They are used in geometry to describe periodic phenomena.

### Special functions

Standard special functions include the sigma function, Euler’s Totient function, the logarithmic integral function, the trigonometric integral, etc.

• Sigma function: For a positive integer ‘n’, the Sigma function is the sum of its positive divisors.

• Euler’s totient function: Euler’s totient function counts positive integers up to a specific integer n that is relatively prime to n. It is also known as Euler’s phi function, written in the Greek letter phi or $\phi$.`

• Logarithmic integral function: This is the integral inverse of logarithms. It is important in the prime numbers’ theorem. The logarithmic integral symbol is $\int$.

• Trigonometric integral: This includes the sine integral, the cosine integral, the hyperbolic sine integral, the hyperbolic cosine integral, Auxiliary functions, and Nielsen’s spiral.

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