Standard Real Functions and Their Graphs

Transcript

0:00Introduction

Hello everyone! So, one quick question. The greatest integer of three point seven is three. Easy, isn't it? Just drop the decimal. Now tell me, what is the greatest integer of minus two point three? Drop the decimal again: minus two, right? Hmm. Most students say that, and lose marks. Hold on to your answer. Today: the standard functions and their graphs. You must draw them in a few seconds, because most domain, range and number of solutions questions in JEE reduce to them. So, let's begin.

0:37Algebraic functions: domain and range

So, first the algebraic functions on screen. The double-struck R means all real numbers, okay? The identity, f of x equals x, and the constant, f of x equals c, accept every real number. The identity gives back every real number; the constant gives only c, in curly brackets. Polynomials also accept all real numbers. For p of x over q of x, remove the points where q of x is zero, because we can never divide by zero. Now, can one by x ever be equal to zero? No, never! So for one by x, zero is missing from the domain, and from the range too. See? The two pieces of the curve never touch either axis. And root x: zero or more goes in, zero or more comes out. So domain and range are both zero to infinity. The square bracket means zero is included; the round bracket means infinity is never reached. Really simple.

1:37Modulus and signum

Alright, now the JEE favourites. Modulus x, with two vertical bars, is x when x is zero or positive, and minus x when x is negative. It is the distance of x from zero, never negative: the V shape on screen. Signum x, written s g n, only reads the sign: one for positive x, zero at zero, and minus one for negative x. On its graph, a filled dot is on the graph, and a hollow dot is not, okay? So modulus x takes every value from zero to infinity, and signum x takes just three values: minus one, zero and one. That's it. Got it? Quick question. The square root of x squared. The square and the root cancel, so it is just x, right? No, no, careful! Classic JEE trap. See, a square root is never negative, but x can be. So the root of x squared is modulus x. And x equals modulus x times signum x: the size, times the sign. Simple as that.

2:42Greatest integer and fractional part

Alright, now the greatest integer of x, written with square brackets. It is the greatest integer less than or equal to x. Some books call it the floor of x, okay? It never goes up. So if x lies between n and n plus one, n included, where n belongs to the double-struck Z, the set of integers, then the greatest integer is n. The little sign like a curved E means belongs to, and the double arrow means implies, okay? That gives the staircase, with a filled dot at the left of each step. Now, our puzzle. Is the greatest integer of minus two point three equal to minus two? Check the number line. Minus two point three lies between minus three and minus two. But minus two is bigger than minus two point three, so it is not allowed! The answer is minus three. Dropping the decimal was the trap; always round down. Now the fractional part, in curly brackets: x, minus the greatest integer of x. It always lies from zero up to one, zero included, one never. That is the saw-tooth on screen. So, the fractional part of minus two point three? Minus two point three, minus of minus three: zero point seven, not zero point three. The fractional part is never negative. Got it?

4:05Two properties you will use again and again

Now, two quick properties. For two point three, the greatest integer is two; for minus two point three, minus three. Sum: minus one. The fractional parts, zero point three and zero point seven, add to one. That holds whenever x is not an integer; the belongs to sign with a line through it means does not belong to. For an integer, both sums are zero. Now, a topper's trick. When an equation mixes x, its greatest integer and its fractional part, write x as the greatest integer plus the fractional part. Then use the box: the fractional part is at least zero and less than one. You will see it in example two.

4:47Exponential and logarithmic functions

Right, the last family: exponential and logarithm, with the base a positive and not equal to one. Can a to the power x ever be zero, or negative? Absolutely never! It is always positive. So the domain is all real numbers, and the range is zero to infinity, with both ends open. The curve comes very close to the x axis, but never touches it. The logarithm is the opposite: domain zero to infinity, range all real numbers. So the log of zero, or of a negative number, is not defined. In every log question, find the domain first, okay? Why opposite? Because they are inverses. Raise a to the power log x, base a, and you get x back. Take the log, base a, of a to the power x: again x. So domain and range just swap places. And the base decides the shape. If a is greater than one, both increase: the blue curves. If a is between zero and one, both decrease: the red curves. Clear?

5:53Example 1: [x] and {x} of 3.7, -3.7, pi and -5

Time for example one. Find the greatest integer and the fractional part of three point seven, and of minus three point seven. Then the greatest integer of pi and of minus pi, and both parts of minus five. Pause the video, and try it first. Done? Okay, let's check. Three point seven: greatest integer three, and fractional part three point seven minus three, that is zero point seven. Easy. Now, minus three point seven lies between minus four and minus three. Round down: minus four, not minus three! Fractional part: minus three point seven, minus of minus four, that is zero point three. Same trap as our puzzle, right? Pi lies between three and four, so its greatest integer is three, and for minus pi it is minus four. Minus five is already an integer: greatest integer minus five, fractional part zero. An integer stays where it is.

6:55Example 2: solve 4{x} = x + [x]

Now a harder one, JEE level. Solve: four times the fractional part of x equals x plus the greatest integer of x. Pause and try. Hint: use the topper's trick. Done? Okay, let's check. Write x as the greatest integer plus the fractional part. Then the right side is two times the greatest integer, plus the fractional part. Cancel one fractional part from both sides: the fractional part equals two times the greatest integer, divided by three. Many students stop here and say: any integer works, so infinitely many solutions, right? No, no, careful! The fractional part lives in its box: at least zero, less than one. So two times the greatest integer must be at least zero and less than three. The greatest integer can only be zero or one. Just two cases! Greatest integer zero: the fractional part is zero, so x is zero. Greatest integer one: the fractional part is two thirds, so x is five thirds. Check: four times two thirds is eight thirds, and five thirds plus one is also eight thirds. Perfectly correct!

8:12Quick revision: graphs, [x] of negatives, root of x^2

So, quick revision. One: picture the graph, and it gives you the domain and the range. Two: the greatest integer rounds down, even for negatives, and the fractional part stays from zero up to one. Three: the root of x squared is modulus x, a to the power x is always positive, and log x needs x positive. Now try the practice problems on this page. Graph first, every time. All the best!