How to Define Functions in Mathematics?


In mathematics, functions serve as the foundation of calculus. Functions are certain forms of relationships. A function is represented as a rule that produces a unique output for each input x. Mathematically, transformation or mapping is used to describe a function. Letters like f, g, and h are commonly used to represent functions. The domain is described as a collection of all possible input values for the function when it is defined. The range refers to all of the values that the function’s output produces. The collection of values that could be outputs of a function is the co-domain. Let’s take a glance at the functions in maths.

What are Functions?

If we state that a variable value y is a function of a variable value x, we mean that y is dependent on x and that y’s value is determined by x’s value. This dependency can be expressed as follows: y = f(x). A function is a method or a relationship that connects each member ‘a’ of a non-empty set A to at least one element ‘b’ of some other non-empty set B. In mathematics, a function is a relation “f” from one set A to another set B. On the other hand, an inverse relation is the inverse of a relation formed by swapping the components of each ordered pair in the provided relation.

In maths, a function is represented as:

  • A collection of ordered pairings
  • Diagram with arrows
  • In table format
  • Graphical representation

Different Types of Functions 

The following are the various categories of functions:

Constant function

A constant function produces the same output value regardless of the input. It’s written as f(x) = c, where c stands for constant. An example of constant function is f(x) = 4.

Polynomial function

A polynomial function is a function that has a polynomial expression. The examples of polynomial function are f(x) = 7x+5, 25x3 – 5x2+ 11x – 4, and so on.

Quadratic function

A quadratic function is one of the types of function with the maximum power 2. The quadratic function is expressed as ax2 + bx + c, and the value of “a” should not be equal to 0. f(x) = 2x2 + 4x + 6 is an example of a quadratic function.

Cubic function

A cubic function is a kind of polynomial function that has the highest power 3 in it. It’s written as f(x) = ax3 + bx2 + cx + d, with the constants a, b, c, and d. Here, a ≠ 0. For example, f(x) = 3x3 + 2x + 5 is a cubic function.

Even and odd function

Let f(x) be a real-valued function. For any value of x in the domain f, a function is said to be an even function if the result of f(-x) is the same as f(x). In comparison, a function is said to be an odd function if the resulting value of f(-x) is the same as the negative of f(x).

Rational function

The ratio of two polynomial functions gives a rational function. It’s written as f(x) = P(x)/Q(x), where P and Q are polynomial functions of variable x and Q(x) ≠ 0. 

Modulus function

A modulus function is a kind of function that determines a number’s absolute value by determining its magnitude. It’s written as f(x) = |x|.

Composite functions

Consider two functions, say f: X → Y and g: Y→ Z. Then, f ∘ g = f(g(x)) for x belongs to X denotes the composition of the function f and g. 

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