MAIN CONCEPTS
We change the variable by letting , rearranging and substituting it into .
Get the Printer Friendly Version

Here is an easy problem to get ourselves acquainted with functions. We shall solve it using a systemic approach and then when we’re familiar with it, we’ll use a short cut.

This is essentially a type of question which requires a ‘change in variable’. What we need is to explicitly find  from a given  where the blank can be any sort of expression written in . In this case, the blank is , fairly easy to manipulate but we’ll go through the full steps. Lets a temporary variable be  such that

and

Substituting this into the function we have

This function is written in terms of . We can simply change it back to  because what is written on the left hand side and right hand side is BOTH in terms of . This gives us

And so

Problem solved. Now for the short cut method. From the , we try to find an expression such that when we substitute into the , we get . This isn’t too difficult to find. We ‘substitute’  into . While this may not be too algebraically correct, it still works. The more correct way is to substitute a dummy variable of say  but I assume the reader knows what I mean. Doing likewise and multiplying by 2, we have

as intended.

This is the short cut in solving this question. However, take note that sometimes functions given in the AHSME aren’t too friendly like in terms of  and so the proper method of the change in variable is desired. There is a question which calls such in my site. Hope you look at it.