Constant

In Mathematics a constant is defined to have a fixed value within the scope of its use. It is usually represented by the first few letters of English alphabet such as a, b, c etc. For example a = 3, c= 7.

Variable

In contrast to constant, a variable can take any value and its value is not fixed. A variable is represented by the last few letters of English alphabet such as x, y, z etc. For example, if you roll a dice, the outcome can be any of the value from set {1, 2, 3, 4, 5, 6}. So, we can say that variable x takes the value of the outcome of a dice. The difference between constant and variable will be clear from the following statement:

            “Add 3 to the outcome of a dice”

We can represent it with the notation of algebra as

            x + a , where x is an element of {1, 2, 3, 4, 5, 6} and a= 3

x can take any value from 1 to 6 and the value of a is fixed within this context.

Function

In order to understand function, we need to know the difference between an independent variable and a dependent variable. Suppose your math teacher gave you 7 homework problems and told that you will be rewarded three times the number of correct solutions. Here, you can solve 1 problem correctly or 2 problems or 3 problems and so on. So, this is an independent variable as it can take any value. On the other hand, your reward, though it is fixed once your correct number of solutions is determined but it is still variable and dependent on the value of correct number of solutions. The relationship that a dependent variable is dependent on independent variable is called function. So, mathematically we can write

Let,        X denotes number of correct answers

                Y denotes reward

X\epsilon \{1, 2, 3, 4, 5, 6, 7\} ; this is read as X is an element of {1, 2, 3, 4, 5, 6, 7}

Y= f(X) this is read as Y is a function of X

=> Y = 3X

The values that X can take is called the domain and the value that are mapped to Y is called range.

Here, domain = \{1,2,3,4,5,6,7\}

           Range = \{3,6,9,12,15,18,21\}

Example: Determine domain and range of f(x) = \sqrt{x-3}

Domain of f(x) is x\geq 3 or \lbrack 3, \infty)

Range of f(x) is \lbrack 0, \infty)

Calculate the value of a function:

f(x) = 7x^2 + 3x + 20 \\

f(1) = 7\times 1^2 + 3\times 1 + 20 \\

f(0) = 7\times 0^2 + 3\times 0 + 20 \\

f(x-7) = 7\times (x-7)^2 + 3\times (x-7) + 20 \\

So, to calculate the value of a function, just substitute value inside the parenthesis of function f() into the right hand side of the describing equation.