## Transforming Functions

Transforming functions is related to the transformation of graphs. A function is basically a sequences of operations on some argument. The argument can be anything, but whatever the argument is, the function does the same sequence of operations, in th same order, to each argument. Suppose that the function f says, 'square the argument'.

We can also write where it is understood that is the argument.

For this example, Suppose now that the argument is not but some other expression in say  says 'square 3x+1 so that This definition of a function is precisely consistent with the transformation of a graph. Continuing with the example above, suppose we replace the argument by so that If we were to transform the graph of to obtain the graph of we would scale the – axis by a factor of When we transform a function however, instead of labelling the x – axis as the x – axis, we label it as the axis. To relabel it the – axis, we have to divide all the – values on the axis by 2 which is the same as scaling by a factor of  In general, when transforming a function by changing the argument, we can just relabel the – axis. To maintain the – axis as the – axis, we are required to perform the inverse sequence of operations that turned into ' ' or ' ' or whatever the new argument is. 