How to approach functional equations problems

A function is the relation between 2 variables. These are important because in most of the situation in life, you find variables. Some are dependent, some are independent. And the relation between them is known as function. For example, y = 5x + 7 is a function. This is an example of explicit function where y is on one side and an expression involving x is on the other side. This is of the form y = f(x).

On the other hand, you have implicit function where the relation between x and y are shown in an indirect way. Some of the implicit functions can be changed to explicit by rearranging the variables while some cannot be changed. For example, x + xy + 2y = 0 is an example of implicit function. If we write the same function in the form of y = -x / (2+x), this becomes explicit. However, functions such as xy = sin y + x2 – y2 cannot be changed into explicit form.

In all these functions, variables are the building blocks.

There are other functions which use functions instead of variables. These are called functional equations. You have come across such functional equations in your 11th/12th while preparing for JEE. These equations are expressed in terms of functions. Some of the most common ones are Cauchy’s functional equations, Jensen’s, and a lot of variants derived out of these. These are the most usual ones you will encounter:

f(x + y) = f(x) + f(y); f(x + y) = f(x).f(y); f(xy) = f(x) + f(y); f(xy) = f(x).f(y). There are more complicated ones a=such as f(x).f(1/x) = f(x) + f(1/x), f((x+y)/2) = (f(x) + f(y))/2 and many more.

You are usually asked to find the function f(x) or the value of the function for a given value of x such as f(2). Here are some of the ways you can approach the problems.

Observation of functional equation

Your syllabus is limited even though it looks like never ending. What is never ending is creativity of JEE question makers which is what makes JEE challenging either by time or by level of difficulty. The function will come from the usual category; these are polynomial with mostly highest degree term with a constant, logarithmic function, exponential, rational, or trigonometric. Looking at the function will give you some insight into the type of function f(x) might be. Some of the questions are dead giveaways and you don’t have to make lot of effort. Some may require more work. For example, sum in the left hand side and product on the right hand side may be a sign of exponential function. You may have to find the multiplication factor and the constant though to get the right answer. These can be found by the additional information given in the question.

Use differentiation by first principle

This “low priority” concept is very useful in functional equation. Using differentiation by first principle gives you the required equation with the term f(x + h). f(x + h) can be used to involve the functional equation given in the question. Using differentiation by first principle shows the relation between the function and its derivative. Once this relation is discovered, you can use integration to find the function.

Use of partial differentiation

The final way is to use partial differentiation. In partial differentiation, either x or y is taken as constant and the whole functional equation is differentiated with respect to the other variable. This means if x is taken as constant, differentiate with respect to y or vice versa. Once this is done, replace one of the variables with a constant. The value of constant depends on the question. If the question doesn’t specifically mentions it, some experiment with 0, 1, -1 etc. should give you enough hint. Once this is done, you get an equation involving function and its derivative. Now use integration to get the right function.

Functional equations can be as difficult and as easy. The scope of variation is humongous and so are the ways of finding the solutions. Many functional equations may still require more work but for your level, these should be enough. You may have to combine these concepts with “theory of equations” concepts in difficult cases though.

~ Pankaj Priyadarshi

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