Simple Functional Equations

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. I give you just one line: f of x, plus two times f of one by x, equals three x. Look, there are two unknowns here, f of x and f of one by x, but only one equation. So, can we still find f? Yes, we can, with one clever substitution! And this very equation came in JEE Main twenty sixteen. So, let's begin.

0:30What is a functional equation?

So, what is a functional equation? It is a rule that a function must satisfy, like our puzzle, okay? And the question asks for the function itself, or just some of its values. Clear? But how do we squeeze more information out of just one rule? See, the rule holds for every allowed x, right? So we can put in clever values: zero, one, minus x, or one by x. Each one gives us a new equation, for free! Then we solve these equations together, like simultaneous equations. That's it, these are the two main tools. Got it?

1:07Clever values in action

Now, let's see tool one in action. Take this rule: f of x plus y equals f of x, plus f of y, for all real x and y. I want f of zero. What should I put in? Yes, put x and y both equal to zero! Then f of zero equals f of zero plus f of zero, that is, two f of zero. So f of zero is zero. As simple as that. Now I want to connect f of x with f of minus x. What should y be? Exactly, y equals minus x! The left side becomes f of zero, which is zero. So f of x plus f of minus x is zero, and f of minus x is minus f of x. So f is an odd function. Perfect. And for a natural number n, put x equals n minus one, and y equals one. Then f of n equals f of n minus one, plus f of one. The three dots on screen mean: keep going down the same way, and you get n times f of one. See? One rule, three facts, just by choosing values smartly.

2:17Standard functional equations

Alright, some equations come again and again, so know their answers directly. First: plus inside the bracket, plus outside. f of x plus y equals f of x, plus f of y. For a continuous f, the answer is f of x equals k x, where k is a constant. Next: plus inside, but times outside. f of x plus y equals f of x, times f of y. Which function turns a sum into a product? Yes, powers! Look, a to the power x plus y equals a to the power x, times a to the power y. So f of x equals a to the power x. But careful, here is a trap. f identically zero also works, because zero is zero times zero. The three lines on screen mean identically equal. Don't miss this answer in the options! Now flip it: times inside, plus outside. Which function does that? The log! Log of x y is log x plus log y. So f of x equals k log x. And times inside, times outside? Now you tell me. Correct, a power! f of x equals x to the power k. Topper's trick: read what happens inside the bracket and what happens outside, and the answer jumps out. With me? Last one, for polynomials: f of x times f of one by x equals f of x, plus f of one by x. The answer is one, plus or minus x to the power n. The data in the question tells you which sign works.

3:52Example 1: f(x) + 2f(1/x) = 3x from JEE Main 2016

Now, back to our puzzle from the start. f of x, plus two f of one by x, equals three x, for x not equal to zero. And S is the set of real x where f of x equals f of minus x. Find this set, okay? Pause the video, and try it first. Done? Okay, let's check. Call the given rule equation one. Two unknowns, so which clever value should we put in? Yes, one by x! The arrow on screen means: replace x by one by x. We get f of one by x, plus two f of x, equals three by x. That is equation two. See, the same two unknowns, but now two equations! Now multiply equation two by two, and subtract equation one. The f of one by x terms cancel, and three f of x equals six by x, minus three x. Divide by three: f of x equals two by x, minus x. Puzzle solved! The whole function, from just one line. Now f of minus x. Two by minus x is minus two by x, and then minus x, right? Then f of x equals f of minus x gives four by x equals zero, so the set is empty? No, no, careful! That is the trap. Minus of minus x is plus x. So f of minus x is minus two by x, plus x. Set the two equal. The double arrow on screen means: if and only if. Bring the terms together, and four by x equals two x. So x squared equals two, and x is plus or minus root two. S has exactly two elements, minus root two and root two. Clear?

5:42Example 2: f(x+y) = f(x)f(y) with a sum of 10 terms

Now a harder one, JEE pattern. f goes from the natural numbers to the real numbers, f of x plus y equals f of x times f of y, and f of one is two. The sum of f of a plus k, for k from one to ten, is sixteen times, two to the power ten, minus one. Find the value of a, okay? Pause and try. Hint: find f of n first. Done? Okay, let's check. f of two is f of one times f of one, and f of three is f of two times f of one. Each step multiplies by f of one again, so f of n is f of one to the power n, that is two to the power n. The a to the power x row from our table! Now the sum. The big Greek sigma on screen means: add these terms. Take two to the power a common. k runs from one to ten, so the bracket is two, plus two squared, and so on, up to two to the power ten. Now this bracket. Is it just two to the power ten, minus one? Many students think so, and get two to the power a equals sixteen, so a equals four. No, no, careful! That is the trap. Our bracket starts at two, not at one. Take two common: two times, one plus two and so on up to two to the power nine, which is two to the power ten, minus one. So the sum is two to the power a plus one, times, two to the power ten, minus one. Compare with sixteen times the same bracket. Two to the power a plus one equals sixteen, that is two to the power four. So a plus one is four, and a equals three. Three, not four! One small factor of two changes the answer.

7:28Quick revision: clever values, systems and standard forms

So, quick revision. One: substitute clever values, zero, one, minus x, or one by x. Two: two unknowns? Make a second equation, then solve the system. Three: know the standard forms, k x, a to the power x, k log x, x to the power k, and one plus or minus x to the power n. Now go and try the practice problems on this page. You will find them easy now, I promise. All the best!