The relationship defined between one independent variable to another dependent variable is called a function. It basically relates an input to an output. The relationship is commonly denoted as y=f(x), which in words is said “f of x”. In a mathematical sense, 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 another non-empty set B. In arithmetic, a function is a relation f from one set A (the domain of the function) to another set B (the co-domain of the function).Therefore a function is denoted as f = {(a,b)| for all a ∈ A, b ∈ B}. 

Types of Function based on Mapping of Elements

We have four functions that are based on the element mapping from set A to set B.

• If the pictures of separate components of A under f are distinct, i.e., for every a, b in A, f(a) = f(b), a = b, then f: A→B is said to be one-to-one or injective. Otherwise, it’s a one-to-many situation.
• f: A→B is said to be onto if every element of B is the image of some element of A under f, i.e. there is an element a in A such that f(a) = b for every b ∈ B. If and only if the function’s range equals B, the function is onto.
• If f is both one-one and onto, it is said to be one-to-one and onto or bijective.

Vertical Line Test of a Function

The term “single-valued” refers to the fact that no vertical line on a graph ever crosses more than one value. It is still a legitimate curve, but not a function if it crosses more than once. The vertical line test determines whether or not a curve is a function. The curve is not a function if it cuts a vertical line at more than one point.

Types of Function on the basis of Graphing

• A polynomial function can be expressed as y=f(a)=f0+f1a+…+fnan, where n∈ N. The graph is always a straight line.
• Linear functions are of the form ax+b=0 where a,b ∈ R, and a are not equal to zero. The graph of such a function will be a straight line.
• Any two functions are called identical functions if the domain of the first function is equal to the domain of the second function and the range of the first function is also equal to the range of the first function.
• Quadratic Functions can be expressed as y=ax2+bx+c amd a,b,c ∈ R, a is not equal to zero. The graph will be a parabola.
• Rational Function are functions of the type f(a)g(a) and g(a) is not equal to zero. Graph of such functions is an Asymptote.
• An algebraic function is a function with a finite number of terms including powers and roots of the independent variable x and fundamental operations like addition, subtraction, multiplication, and division.
• Cubic Function is the polynomial function of degree 3.
• Modulus Function is defined as f(a)=|a|=a, a≥0 and f(a)= -a, a<0. 
• Signum Function is also called the sign function, which gives the sign of real number. The domain is the where the function is defined and range is always {-1,0,1}
• Greatest Integer Function is defined as f(a)=[a] gives the value of the greatest integer less than or equal to a. It is always an integral output. Its domain is R and Range is integers.
• Fractional Part Function gives fractional value as an output. 
• Even an Odd Function– If f(x) = f(-x), the function is an even one, and if f(x) = -f(-x), the function is an odd one.
• If there exists a positive real integer T such that f(u – t) = f(x) for all x∈  Domain, the function is said to be a periodic function.
• Composite Function– Let f and g be two functions, then the composition of f and g is denoted as f(g) and is defined as fog=f(g(x)).
• Constant Function is the polynomial function with degree 0. The graph is a straight line parallel to x-axis.
• Identity Function is defined as b=f(a)=a. The graph of such a function is a straight line passing through the origin.

Algebra of Functions

The algebra of functions is concerned with function operations. We have the following for the functions f(x) and g(x), where f: X→ R and g: X →R, respectively, and x∈ X:

• (f-g)(x)=f(x)+g(x)
• (f+g)(x)=f(x)+g(x)
• f.g(x)=f(x).g(x)
• (k.f(x))=k(f(x)), where k is any real number
• (f/g)(x)=f(x)/g(x), where g(x) cannot be equal to zero.