One-one and Many-one Functions

Transcript

0:00Introduction

Hello everyone! So, one small question before we start. In your class, every student has exactly one roll number, and exactly one birthday, okay? So both are functions: one student in, one answer out. Now tell me. Can two students share a roll number? Never, right? And can two students share a birthday? Yes, easily! So both are functions, but they behave very differently. Today we name this difference: one-one, and many-one. And look, in JEE the question, is this function one-one, comes again and again in this chapter, right up to inverse functions. So, let's begin.

0:42One-one (injective) and many-one

So, let's write our roll number idea in maths. Take a function f, from set A to set B, okay? We call f one-one if different inputs always give different outputs. Just like roll numbers: two different students, two different roll numbers. The fancy name is injective, okay? In symbols: x one not equal to x two, implies, f of x one not equal to f of x two. The equals sign with a slash through it means not equal to, and the double arrow means implies, okay? Different in, different out. Clear? But wait. On the real numbers, there are endless pairs of different inputs. How will you check every single pair? Here is the trick: we turn it around! Start from equal outputs instead. If f of x one equals f of x two, then x one must equal x two. Same meaning, but much easier to use: just one equation to solve, okay? And the opposite? If two different inputs share an output, f is many-one. Like birthdays: two different students, same birthday. Just one such pair is enough.

1:54Tests 1 and 2: algebra, or one counterexample

Now, how do we actually test a function? There are four ways. Test one, algebra. Assume f of x one equals f of x two, and work until you reach x one equals x two. If you get there, the function is one-one, okay? Test two, a counterexample. Find two different inputs, x one and x two, that give the same output. Found them? Then f is many-one, done. Look, if you suspect many-one, skip the algebra and hunt for one pair.

2:25Test 3: the horizontal line test

Test three is for graphs. Remember the vertical line test, which tells us if a graph is a function? Now we turn the line sideways, and use horizontal lines. f is one-one, if and only if, each horizontal line... The two-sided arrow on screen means if and only if, okay? meets the graph at most once. Once, or not at all, but never twice. Got it? Look at the left graph, y equals x squared, the parabola. How many times does the dashed line cut it? Twice! Exactly. Two different values of x, same height, so the same output. So x squared fails the test on the real numbers: many-one. The double-struck R on screen means the set of real numbers, okay? Now the right graph, y equals x cubed. Any horizontal line you draw cuts it only once. So x cubed passes: one-one. Crystal clear.

3:27Test 4: always rising, or always falling

Test four. Imagine a graph that only ever goes up, strictly increasing. Can it ever come back to the same height? Never, right? So no output can repeat. One-one! Same for strictly decreasing, always going down. And if f is differentiable on an interval, look at its derivative, f dash of x, the slope. If f dash of x is greater than zero everywhere, or less than zero everywhere, f is one-one. With me? One small detail. If this fails only at isolated points, single separate points, the function is still one-one, okay? Like x cubed: its slope, three x squared, is zero only at x equals zero. Just one point, so x cubed is still one-one, exactly what the graph showed. Clear?

4:20Topper's trick: spot many-one at a glance

Now a trick that toppers use. Some functions, you can call many-one in one second. First, even functions, where f of minus x equals f of x. Why are they many-one? Because f of minus one equals f of one, and minus one is not one! That's it, counterexample ready. Second, periodic functions. Their values repeat again and again, so the same output comes back many times. Many-one. Third, any continuous function that goes up and then comes down, or has a turning point. Near the turning point, a horizontal line cuts the graph on both sides, right? Many-one. And fourth: a polynomial of even degree is always many-one on the real numbers. Like x squared, the parabola we just saw. Now careful. If even degree means many-one, then odd degree means one-one, right? No, no! This is the JEE trap. The trick says nothing about odd degree. For odd degree, you must actually test. You will see this in example two, okay?

5:29Example 1: test 2x+3, x^2 on two domains and a pair set

Now, example one. Test each of these for one-one. First, f from R to R, that is, from the real numbers to the real numbers: f of x equals two x plus three. Second, g of x equals x squared, again from R to R, okay? Third, h of x equals x squared again, but only from zero to infinity. The square bracket means zero is included, okay? And fourth, a function given as pairs: one goes to letter a, two goes to letter b, and three goes to letter a again. Pause the video, and try it first. Done? Okay, let's check. Part one, algebra. Assume two x one plus three equals two x two plus three. Subtract three, divide by two: x one equals x two. One-one! Part two: g of x equals x squared, on the real numbers. Counterexample: g of minus one equals g of one, both are one. Many-one, just like the parabola in our graph. Alright, part three is x squared again. So, many-one again, right? No, no, careful! Look at the domain: only zero and above. If x one squared equals x two squared, and both are greater than or equal to zero, then x one equals x two. One-one! Minus one is simply not in the domain. So remember: the domain matters! And part four. Both one and three go to letter a, the same output. Two students, same birthday! Many-one.

7:09Example 2: is x^3+x, x^3-x, |x|+x or e^x+e^-x one-one?

Now a harder one, JEE level. Test each of these on the real numbers. One: x cubed plus x. Two: x cubed minus x. Three: modulus of x, plus x. The two straight bars mean modulus, okay? And four: e to the x, plus e to the minus x. Pause and try. Hint: pick the right test for each one. Done? Okay, let's check. Part one, x cubed plus x. Use the slope: f dash of x is three x squared plus one. Now, x squared is never negative, so this is always greater than zero. Strictly increasing, so one-one! Part two, x cubed minus x. Also degree three, an odd degree. So one-one too, right? Careful, remember the trap! Odd degree tells you nothing. Try zero, one and minus one. Zero minus zero is zero, one minus one is zero, and minus one plus one is also zero. Three different inputs, one output. Many-one! Part three, modulus of x, plus x. For any x less than or equal to zero, modulus of x is minus x, so the sum is zero. Every such x gives zero! Many-one. And part four, e to the x plus e to the minus x. Put minus x in place of x, and you get the same expression back. So it is even, and our topper's trick says: many-one. Indeed, f of one equals f of minus one. Done in one line. Got it?

8:45Quick revision: definition, counterexample, line test

So, quick revision. One-one means equal outputs force equal inputs. For many-one, one counterexample is enough. Use the horizontal line test or the slope, and always, always check the domain. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!