**Graph of a function**

It is a plot of values of function against independent variable x. The value of function changes in accordance with function rule as x changes. Graph depicts these changes pictorially. In the current context, both core function and modified function graphs are plotted against same independent variable x.

**Input to the function**

What is input to the function? How do we change input to the function? Values are passed to the function through argument of the function. The argument itself is a function in x i.e. independent variable. The simplest form of argument is "x" like in function f(x). The modified arguments are "2x" in function f(2x) or "2x-1" in function f(2x-1). This changes input to the function. Important to underline is that independent variable x remains what it is, but argument of the function changes due to mathematical operation on independent variable. Thus, we modify argument though mathematical operation on independent variable x. Basic possibilities of modifying argument i.e. input by using arithmetic operations on x are addition, subtraction, multiplication, division and negation. In notation, we write modification to the input of the function as :

These changes are called internal or pre-composition modifications.

**Output of the function**

A modification in input to the graph is reflected in the values of the function. This is one way of modifying output and hence corresponding graph. Yet another approach of changing output is by applying arithmetic operations on the function itself. We shall represent such arithmetic operations on the function as :

These changes are called external or post-composition modifications.