If AoPS has SFFT, we also have Joselito’s Clay Molding Technique ( JCMT ). Before anyone will name this technique, I decided to publish this and name it after the person who popularized the method.

JCMT is a technique first used by Engr. Joselito Torculas, one of the admins of Elite Math Circle. The technique is used in composition of functions. He described the method as “*shaping the clay to its form*”. Thus the name is JCMT.

Illustrative Examples:

Worked Example 1:

Let and . What is ?

Solution: By JCMT,

We know that

But ,

But ,

Now, since we are just looking for , we need to look for a value of x to make 2x-1 become x. This can be easily accomplished using the inverse of 2x-1 but we will not use that.

The first stage of JCMT is to eliminate the constant in 2x-1 and eventually x and to the desired function.

Let

Let

Worked Example 2:

Given that . What is the value of

Solution: By JCMT,

Let

Let

Finally, let

Worked Example 3:

If . What is

Solution: By JCMT

Let

Let

for example in #2, why not equate directly x=(x+4)/2 to get f(x), same as to #3, equate directly x=(x-1)/2 to get f(x)? Like for #3, if you will equate it x=(x-1)/2 directly to the RHS, f(x)=3[(x-1)/2]^2 -1 = (3x^2 -6x -1)/4 so, f(x)=(3x^4 -6x^2 -1)/4 and it will yield the same answer as yours…

Right! But not all people can figure that out especially to those who new to this. 🙂 One step at a time. Thanks John! 🙂

Minor comment: On Example 1, the dot sign (.) was incorrectly used to denote composition of functions. Dot usually means multiplication. The symbol used for composition is usually a circle (circ in latex, IIRC).

Otherwise, very good method. I usually just use the inverse of the function inside the composition, but I can see how this could be easier and more elegant in some cases. Keep it up!