1. Definition of function

Function is a relation whereby every element in the domain has one unique image in the range. There are four types of relations:

• one-to-one
• one-to-many
• many-to-one
• many-to-many

Based on the definition, one-to-one relation and many-to-one relation are functions. Many-to-one relation and many-to-many relation are not functions.

2. Notation of function

The common way of writing a function is

where 

x is the independent variable, y is the dependent variable and f is the function name. 

The independent variable x is sometimes called the input and the dependent variable y is sometimes called the output. 

It is also common to write the function without a function name. For instance, you can write as y=fx=x or as y=x.

Example 

y=f(x)=x is a one-to-one function because every element of x has a unique value of y. For example, when x=-2, y=-2, when x=3, y=3, when  \(x={1\over2}\)  and  \(y={1\over2}\)  and so on. 

\(y=f(x)=x^2 \)   is a many-to-one function because two different values of  x can be mapped onto the same y. For example, when x= -2,  \(y=(-2)^2\) ,  y=-2=4 and when x=2, \(y=(2)^2=4 \)  and that is  f-2=f2=4.

3. Domain and range

Domain of a function is the set of all the values of the independent variable, 

x for which the function is defined. Range of a function is the set of all the values of the dependent variable, y that corresponds to the domain. You can use capital D to denote domain and capital R to denote range of a function.

Example

1. The domain for y=f(x)=x is all real numbers of x, written as  D: x∈R. Since y=x, the range is also all real numbers, written as  R: y∈R.

2. The domain for  y=f(x)=x2 is all real numbers because any real number can be squared. The range consists of y=0 and all positive values of y because the square of any number cannot be negative. Note that the square of 0 is still 0. Thus, domain D:x∈R and R: y≥0.

4. Inverse functions

Inverse of function 

f(x), written as \(f^{-1}(x)\), is obtained by interchanging x and y.

Example 

1. Find the inverse function for the following functions:y=f(x)=x+2

Interchanging x and y in function y=x+2, you get x=y+2. Then, y=x-2.

∴  \(f-^{1}(x)=x-2\)

2. y=f(x)=2x-5

Interchanging x and y in function y=2x-5, you get x=2y-5.

Then, 2y=x+5 and  \(y={1\over2}(x+5)\)

∴  \(f^{-1}x={1\over2}(x+5) \)

3. \(y=f(x)={x\over{x-2}}\)

Interchanging x and y in function \(y={x\over{x-2}}\) you get \(x={y\over{y-2}}\). 

Then, 

          xy-2x=y 

          xy-y=2x

          y(x-1)=2x

          \(y={2x\over{x-1}}\)

∴  \(f^{-1}(x)={2x\over{(x-1)}}\)