## A little portrait of a genius mathematician

Reference: A Beautiful Mind by Sylvia Nasar, the life of mathematical genius and Nobel Laureate, John Nash, A Touchstone Book, Published by Simon and Schuster.

…”Geniuses”, the mathematician Paul Halmos wrote, “are of two kinds: the ones who are just like all of us, but very much more so, and the ones who apparently have an extra human spark. We can all run, and some of us can run the mile in less than 4 minutes, but there is nothing that most of us can do that compares with the creation of the Great G-minor Fugue.” Nash’s genius was of that mysterious variety more often associated with music and art than with the oldest of all sciences. It wasn’t that his mind worked faster, that his memory was more retentive or that his power of concentration was greater. The flashes of intuition were non-rational. Like other great mathematical intuitionists — Georg Friedrich Bernhard Riemann, Jules Henri Poincare, Srinivasa Ramanujan —- Nash saw the vision first constructing the laborious proofs long afterwards. But even after he would try to explain some astonishing result, the actual route he had taken remained a mystery to others who tried to follow his reasoning. Donald Newman, a mathematician who knew Nash at MIT in the 1950s, used to say about him that “everyone else would climb a peak by looking for a path somewhere on the mountains. Nash would climb another mountain altogether and from that distant peak would shine a searchlight back onto the first peak.”

Hats off,

Nalin Pithwa.

## A little portrait of Hermann Weyl

“A Proteus who transforms himself ceaselessly in order to elude the grip of his adversary, not becoming himself again until after the final victory.” Thus, Hermann Weyl (1885-1955) appeared to his eminent younger colleagues Claude Chevalley and Andre Weil. Surprising words to describe a mathematician, but apt for the amazing variety of shapes and forms in which Weyl’s extraordinary abilities revealed themselves, for “among all the mathematicians who began their working life in the twentieth century, Hermann Weyl was the one who made major contributions in the greatest number of different fields. He alone could stand comparison with the last great universal mathematicians of the nineteenth century, David Hilbert and Henri Poincare,” in the view of Freeman Dyson. “He was indeed not only a great mathematician but a great mathematical writer,” wrote another colleague.

https://www.amazon.in/Concept-Riemann-Surface-Hermann-Weyl/dp/160796239X/ref=sr_1_1_sspa?keywords=Hermann+Weyl&qid=1574319421&s=books&sr=1-1-spons&psc=1&spLa=ZW5jcnlwdGVkUXVhbGlmaWVyPUEySUUzOVBXM0ZHTzZPJmVuY3J5cHRlZElkPUEwODY2MzM5TUUzMzlCVVRYREpRJmVuY3J5cHRlZEFkSWQ9QTAwNjQ5MDJXUk03SkhIQk1HRE8md2lkZ2V0TmFtZT1zcF9hdGYmYWN0aW9uPWNsaWNrUmVkaXJlY3QmZG9Ob3RMb2dDbGljaz10cnVl

https://www.amazon.in/Continuum-Critical-Examination-Foundation-Mathematics/dp/0486679829/ref=sr_1_17?keywords=Hermann+Weyl&qid=1574319421&s=books&sr=1-17

Regards

Nalin Pithwa

## Theory of Quadratic Equations: Part III: Tutorial practice problems: IITJEE Mains and preRMO

Problem 1:

Find the condition that a quadratic function of x and y may be resolved into two linear factors. For instance, a general form of such a function would be : $ax^{2}+2hxy+by^{2}+2gx+2fy+c$.

Problem 2:

Find the condition that the equations $ax^{2}+bx+c=0$ and $a^{'}x^{2}+b^{'}x+c^{'}=0$ may have a common root.

Using the above result, find the condition that the two quadratic functions $ax^{2}+bxy+cy^{2}$ and $a^{'}x^{2}+b^{'}xy+c^{'}y^{2}$ may have a common linear factor.

Problem 3:

For what values of m will the expression $y^{2}+2xy+2x+my-3$ be capable of resolution into two rational factors?

Problem 4:

Find the values of m which will make $2x^{2}+mxy+3y^{2}-5y-2$ equivalent to the product of two linear factors.

Problem 5:

Show that the expression $A(x^{2}-y^{2})-xy(B-C)$ always admits of two real linear factors.

Problem 6:

If the equations $x^{2}+px+q=0$ and $x^{2}+p^{'}x+q^{'}=0$ have a common root, show that it must be equal to $\frac{pq^{'}-p^{'}q}{q-q^{'}}$ or $\frac{q-q^{'}}{p^{'}-p}$.

Problem 7:

Find the condition that the expression $lx^{2}+mxy+ny^{2}$ and $l^{'}x^{2}+m^{'}xy+n^{'}y^{2}$ may have a common linear factor.

Problem 8:

If the expression $3x^{2}+2Pxy+2y^{2}+2ax-4y+1$ can be resolved into linear factors, prove that P must be be one of the roots of the equation $P^{2}+4aP+2a^{2}+6=0$.

Problem 9:

Find the condition that the expressions $ax^{2}+2hxy+by^{2}$ and $a^{'}x^{2}+2h^{'}xy+b^{'}y^{2}$ may be respectively divisible by factors of the form $y-mx$ and $my+x$.

Problem 10:

Prove that the equation $x^{2}-3xy+2y^{2}-2x-3y-35=0$ for every real value of x, there is a real value of y, and for every real value of y, there is a real value of x.

Problem 11:

If x and y are two real quantities connected by the equation $9x^{2}+2xy+y^{2}-92x-20y+244=0$, then will x lie between 3 and 6, and y between 1 and 10.

Problem 11:

If $(ax^{2}+bx+c)y+a^{'}x^{2}+b^{'}x+c^{'}=0$, find the condition that x may be a rational function of y.

More later,

Regards,

Nalin Pithwa.

## Theory of Quadratic Equations: part II: tutorial problems: IITJEE Mains, preRMO

Problem 1:

If x is a real number, prove that the rational function $\frac{x^{2}+2x-11}{2(x-3)}$ can have all numerical values except such as lie between 2 and 6. In other words, find the range of this rational function. (the domain of this rational function is all real numbers except $x=3$ quite obviously.

Problem 2:

For all real values of x, prove that the quadratic function $y=f(x)=ax^{2}+bx+c$ has the same sign as a, except when the roots of the quadratic equation $ax^{2}+bx+c=0$ are real and unequal, and x has a value lying between them. This is a very useful famous classic result.

Remarks:

a) From your proof, you can conclude the following also: The expression $ax^{2}+bx+c$ will always have the same sign, whatever real value x may have, provided that $b^{2}-4ac$ is negative or zero; and if this condition is satisfied, the expression is positive, or negative accordingly as a is positive or negative.

b) From your proof, and using the above conclusion, you can also conclude the following: Conversely, in order that the expression $ax^{2}+bx+c$ may be always positive, $b^{2}-4ac$ must be negative or zero; and, a must be positive; and, in order that $ax^{2}+bx+c$ may be always negative, $b^{2}-4ac$ must be negative or zero, and a must be negative.

Further Remarks:

Please note that the function $y=f(x)=ax^{2}+bx+c$, where $a, b, c \in \Re$ and $a \neq 0$ is a parabola. The roots of this $y=f(x)=0$ are the points where the parabola cuts the y axis. Can you find the vertex of this parabola? Compare the graph of the elementary parabola $y=x^{2}$, with the graph of $y=ax^{2}$ where $a \neq 0$ and further with the graph of the general parabola $y=ax^{2}+bx+c$. Note you will just have to convert the expression $ax^{2}+bx+c$ to a perfect square form.

Problem 3:

Find the limits between which a must lie in order that the rational function $\frac{ax^{2}-7x+5}{5x^{2}-7x+a}$ may be real, if x is real.

Problem 4:

Determine the limits between which n must lie in order that the equation $2ax(ax+nc)+(n^{2}-2)c^{2}=0$ may have real roots.

Problem 5:

If x be real, prove that $\frac{x}{x^{2}-5x+9}$ must lie between 1 and $-\frac{1}{11}$.

Problem 6:

Prove that the range of the rational function $y=f(x)=\frac{x^{2}-x+1}{x^{2}+x+1}$ lies between 3 and $\frac{1}{3}$ for all real values of x.

Problem 7:

If $x \in \Re$, Prove that the rational function $y=f(x)=\frac{x^{2}+34x-71}{x^{2}+2x-7}$ can have no value between 5 and 9. In other words, prove that the range of the function is $(x <5)\bigcup(x>9)$.

Problem 8:

Find the equation whose roots are $\frac{\sqrt{a}}{\sqrt{a} \pm \sqrt(a-b)}$.

Problem 9:

If $\alpha, \beta$ are roots of the quadratic equation $x^{2}-px+q=0$, find the value of (a) $\alpha^{2}(\alpha^{2}\beta^{-1}-\beta)+\beta^{2}(\beta^{2}\alpha^{-1}-\alpha)$ (b) $(\alpha-p)^{-4}+(\beta-p)^{-4}$.

Problem 10:

If the roots of $lx^{2}+mx+n=0$ be in the ratio p:q, prove that $\sqrt{\frac{p}{q}}+\sqrt{\frac{q}{p}}+\sqrt{\frac{n}{l}}=0$

Problem 11:

If x be real, the expression $\frac{(x+m)^{2}-4mn}{2(x-n)}$ admits of all values except such as those that lie between 2n and 2m.

Problem 12:

If the roots of the equation $ax^{2}+2bx+c=0$ are $\alpha$ and $\beta$, and those of the equation $Ax^{2}+2Bx+C=0$ be $\alpha+\delta$ and $\beta+\delta$, prove that $\frac{b^{2}-ac}{a^{2}} = \frac{B^{2}-AC}{A^{2}}$.

Problem 13:

Prove that the rational function $y=f(x)=\frac{px^{2}+3x-4}{p+3x-4x^{2}}$ will be capable of all values when x is real, provided that p has any real value between 1 and 7. That is, under the conditions on p, we have to show that the given rational function has as its range the full real numbers. (Of course, the domain is real except those values of x for which the denominator is zero).

Problem 14:

Find the greatest value of $\frac{x+2}{2x^{2}+3x+6}$ for any real value of x. (Remarks: this is maxima-minima problem which can be solved with algebra only, calculus is not needed).

Problem 15:

Show that if x is real, the expression $(x^{2}-bc)(2x-b-c)^{-1}$ has no real value between b and a.

Problem 16:

If the roots of $ax^{2}+bx+c=0$ be possible (real) and different, then the roots of $(a+c)(ax^{2}+2bx+c)=2(ac-b^{2})(x^{2}+1)$ will not be real, and vice-versa. Prove this.

Problem 17:

Prove that the rational function $y=f(x)=\frac{(ax-b)(dx-c)}{(bx-a)(cx-a)}$ will be capable of all real values when x is real, if $a^{2}-b^{2}$ and $c^{2}-a^{2}$ have the same sign.

Cheers,

Nalin Pithwa

## Theory of Quadratic Equations: Tutorial problems : Part I: IITJEE Mains, preRMO

I) Form the equations whose roots are:

a) $-\frac{4}{5}, \frac{3}{7}$ (b) $\frac{m}{n}, -\frac{n}{m}$ (c) $\frac{p-q}{p+q}, -\frac{p+q}{p-q}$ (d) $7 \pm 2\sqrt{5}$ (e) $-p \pm 2\sqrt{2q}$ (f) $-3 \pm 5i$ (g) $-a \pm ib$ (h) $\pm i(a-b)$ (i) $-3, \frac{2}{3}, \frac{1}{2}$ (j) $\frac{a}{2},0, -\frac{2}{a}$ (k) $2 \pm \sqrt{3}, 4$

II) Prove that the roots of the following equations are real:

i) $x^{2}-2ax+a^{2}-b^{2}-c^{2}=0$

ii) $(a-b+c)x^{2}+4(a-b)x+(a-b-c)=0$

III) If the equation $x^{2}-15-m(2x-8)=0$ has equal roots, find the values of m.

IV) For what values of m will the equation $x^{2}-2x(1+3m)+7(3+2m)=0$ have equal roots?

V) For what value of m will the equation $\frac{x^{2}-bx}{ax-c} = \frac{m-1}{m+1}$ have roots equal in magnitude but opposite in sign?

VI) Prove that the roots of the following equations are rational:

(i) $(a+c-b)x^{2}+2ax+(b+c-a)=0$

(ii) $abc^{2}x^{2}+3a^{2}cx+b^{2}ax-6a^{2}-ab+2b^{2}=0$

VII) If $\alpha, \beta$ are the roots of the equation $ax^{2}+bx+c=0$, find the values of

(i) $\frac{1}{\alpha^{2}} + \frac{1}{\beta^{2}}$

(ii) $\alpha^{4}\beta^{7}+\alpha^{7}\beta^{4}$

(iii) $(\frac{\alpha}{\beta}-\frac{\beta}{\alpha})^{2}$

VIII) Find the value of:

(a) $x^{3}+x^{2}-x+22$ when $x=1+2i$

(b) $x^{3}-3x^{2}-8x+16$ when $x=3+i$

(c) $x^{3}-ax^{2}+2a^{2}x+4a^{3}$ when $\frac{x}{a}=1-\sqrt{-3}$

IX) If $\alpha$ and $\beta$ are the roots of $x^{2}+px+q=0$ form the equation whose roots are $(\alpha-\beta)^{2}$ and $(\alpha+\beta)^{2}$/

X) Prove that the roots of $(x-a)(x-b)=k^{2}$ are always real.

XI) If $\alpha_{1}, \alpha_{2}$ are the roots of $ax^{2}+bx+c=0$, find the value of (i) $(ax_{1}+b)^{-2}+(ax_{2}+b)^{-2}$ (ii) $(ax_{1}+b)^{-3}+(ax_{2}+b)^{-3}$

XII) Find the condition that one root of $ax^{2}+bx+c=0$ shall be n times the other.

XIII) If $\alpha, \beta$ are the roots of $ax^{2}+bx+c=0$ form the equation whose roots are $\alpha^{2}+\beta^{2}$ and $\alpha^{-2}+\beta^{-2}$.

XIV) Form the equation whose roots are the squares of the sum and of the differences of the roots of $2x^{2}+2(m+n)x+m^{2}+n^{2}=0$.

XV) Discuss the signs of the roots of the equation $px^{2}+qx+r=0$

XVI) If a, b and c are odd integers, prove that the roots of the equation $ax^{2}+bx+c=0$ cannot be rational numbers.

XVII) Given that the equation $x^{4}+px^{3}+qx^{2}+rx+s=0$ has four real positive roots, prove that (a) $pr-16s \geq 0$ (b) $q^{2}-36s \geq 0$, where equality holds, in each case, if and only if the roots are equal.

XVIII) Let $p(x)=x^{2}+ax+b$ be a quadratic polynomial in which a and b are integers. Given any integer n, show that there is an integer M such that $p(n)p(n+1)=p(M)$.

Cheers,

Nalin Pithwa.

## Set theory, functions, relations: part VI

What follows are some more practice questions on functions. The questions are not challenging but we can say that they do lead to conceptual clarity and present some standard set of questions on this topic (it behooves every beginner in calculus or IITJEE mains or RMO or pre RMO to try these set of questions):

1. Find the domain and range of the function: $f(x)=\frac{x-2}{3-x}$
2. If $f(x)=3x^{3}-5x^{2}+9$, find $f(x-1)$.
3. If $f(x)=x^{3}-\frac{1}{x^{3}}$, show that $f(x)+f(\frac{1}{x})=0$
4. If $f(x)=\frac{x+1}{x-1}$ show that $f(f(x))=x$.
5. Find the domain and range of the real valued function $f(x)=\frac{x^{2}+2x+1}{x^{2}-8x+12}$
6. Find the domain of the real valued function of a real variable: $f(x)=\frac{x-2}{2-x}$
7. Find the domain and range of the real valued function $f(x)=\frac{1}{1-x^{2}}$.
8. A function $f: \Re \longrightarrow \Re$ is defined by $f(x)=\frac{3x}{5}+2$ where $x \in \Re$. Does the inverse of f exist? If so, find it. Also, find the domain and range of the inverse.
9. A function is defined piece-wise as follows: $f(x)=3x+5$ for $- 4 \leq x \leq 0$ and $f(x)=5-3x$ for $0 < x \leq 4$, find $f(f(\frac{5}{2}))$; the domain and range of f; and the value of x for which $f(x)=-4$
10. If $f: \Re \longrightarrow \Re$ and $g: \Re \longrightarrow \Re$ given by $f(x)=x-5$ and $g(x)=x^{2}-1$, find (a) $f \circ g$ (b) $g \circ f$ (c) $f \circ f$ and (d) $g \circ g$
11. Find $f(g(x))$ and $g(f(x))$ if (a) $f(x)=3x-1$ and $g(x)=x^{2}+1$ (b) $f(x)=2x$ and $g(x)=4x+1$
12. If $f(x)=\frac{3x+4}{5x-7}$ and $g(x)=\frac{7x+4}{5x-3}$ prove that $f(g(x))=g(f(x))=x$
13. Find the domain and range of the following functions: (a) $f(x)=x^{2}$ (b) $f(x)=\sqrt{(x-1)(3-x)}$ (c) $f(x)=\frac{1}{\sqrt{x^{2}-1}}$ (d) $f(x)=\frac{x+3}{x-3}$ (e) $f(x)=\sqrt{9-x^{2}}$ (f) $f(x)=\sqrt{\frac{x-2}{3-x}}$
14. Find the range of each of the following functions: (a) $f(x)=3x-4$, when $-1 \leq x <3$ (b) $f(x)=9-2x^{2}$ for $-5 \leq x \leq 3$ (c) $f(x)=x^{2}-6x+11$ for all $x \in \Re$.
15. Solve the following: (a) if $f(x)=\frac{x^{3}+1}{x^{2}+1}$, find $f(-3)$, and $f(-1)$. (b) If $f(x)=(x-1)(2x+1)$, find $f(1)$, $f(2)$, $f(-3)$. (c) If $f(x)=2x^{2}-3x-1$, find $f(x+2)$.
16. Which of the following relations are functions? Justify your answer. If it is a function, determine its range and domain. (a) $\{ (2,1),(4,2),(6,3),(8,4), (10,5), (12,6),(14,7)\}$ (b) $\{ (2,1),(3,1),(5,2)\}$ (c) $\{ (2,3),(3,2),(2,5),(5,2)\}$ (d) $\{ (0,0),(1,1),(1,-1),(4,2),(4,-2),(9,3),(9,-3),(16,4),(16,-4)\}$
17. Find a, if $f(x)=ax+5$, and $f(1)=8$
18. If $f(x)=f(3x-1)$ for $f(x)=x^{2}-4x+11$, find x.
19. If $f(x)=x^{2}-3x+4$, then find the value of x satisfying $f(x)=f(2x+1)$.
20. Let $A = \{ 1,2,3,4 \}$ and $Z$ be the set of integers. Define $f:A \longrightarrow Z$ by $f(x)=3x+7$. Show that f is a function from A to Z. Also, find the range of f.
21. Find if the following functions are one-one or onto or bijective: (a) $f: \Re \longrightarrow \Re$ (b) $f: Z \longrightarrow Z$ given by $f(x)=x^{2}+4$ for all $x \in Z$.
22. Find which of the following functions are surjective, injective or bijective or none of these : (a) $f: \Re \longrightarrow \Re$ as $f(x)=3x+7$ for all $x \in \Re$ (b) $f: \Re \longrightarrow \Re$ given as $f(x)=x^{2}$ for all $x \in \Re$ (c) $f = \{ (1,3),(2,6),(3,9),(4,12)\}$ defined from A to B where $A = \{ 1,2,3,4\}$ and $B = \{ 5,6,9,12,15\}$
23. Let f and g be two real valued functions defined by $f(x)=x+1$ and $g(x)=2x-9$. Find $f+g$ and $f-g$ and $\frac{f}{g}$.
24. Find $g \circ f$ and $f \circ g$ where (a) $f(x)=x-2$ and $g(x)=x^{2}+3x+1$ (b) $f(x)=\frac{1}{x}$ and $g(x)=\frac{x-2}{x+2}$.
25. If $f(x)= \frac{2x+3}{3x-2}$ prove that $f \circ f$ is an identity function.
26. If $f(x)=\frac{3x+2}{4x-1}$ and $g(x)=\frac{x+2}{4x-3}$, prove that $(g \circ f)(x)=(f \circ g)(x)=x$.
27. If $f = \{ (2,4),(3,6),(4,8),(5,10),(6,12)\}$ and $g = \{ (4,13),(6,19),(8,25),(10,31),(12,37)\}$ find $g \circ f$.
28. Show that $f:\Re \longrightarrow \Re$ given by $f(x)=3x-4$ is one-one and onto also. Find its inverse function also. Also, find the domain and range of the inverse function. Also find $f^{-1}(9)$ and $f^{-1}(-2)$
29. Let $f: \Re-\{ 2\} \longrightarrow \Re$ be defined by $f(x)=\frac{x^{2}-4}{x-2}$ and $g: \Re \longrightarrow \Re$ be defined by $g(x)=x+2$. Find whether the two functions f and g are same, or not same. Justify your answers.

Regards,

Nalin Pithwa

## Set theory, relations, functions: preliminaries: Part V

Types of functions: (please plot as many functions as possible from the list below; as suggested in an earlier blog, please use a TI graphing calculator or GeoGebra freeware graphing software):

1. Constant function: A function $f:\Re \longrightarrow \Re$ given by $f(x)=k$, where $k \in \Re$ is a constant. It is a horizontal line on the XY-plane.
2. Identity function: A function $f: \Re \longrightarrow \Re$ given by $f(x)=x$. It maps a real value x back to itself. It is a straight line passing through origin at an angle 45 degrees to the positive X axis.
3. One-one or injective function: If different inputs give rise to different outputs, the function is said to be injective or one-one. That is, if $f: A \longrightarrow B$, where set A is domain and set B is co-domain, if further, $x_{1}, x_{2} \in A$ such that $x_{1} \neq x_{2}$, then it follows that $f(x_{1}) \neq f(x_{2})$. Sometimes, to prove that a function is injective, we can prove the conrapositive statement of the definition also; that is, $y_{1}=y_{2}$ where $y_{1}, y_{2} \in codomain \hspace{0.1in} range$, then it follows that $x_{1}=x_{2}$. It might be easier to prove the contrapositive. It would be illuminating to construct your own pictorial examples of such a function.
4. Onto or surjective: If a function is given by $f: X \longrightarrow Y$ such that $f(X)=Y$, that is, the images of all the elements of the domain is full of set Y. In other words, in such a case, the range is equal to co-domain. it would be illuminating to construct your own pictorial examples of  such a function.
5. Bijective function or one-one onto correspondence: A function which is both one-one and onto is called a bijective function. (It is both injective and surjective). Only a bijective function will have a well-defined inverse function. Think why! This is the reason why inverse circular functions (that is, inverse trigonometric functions have their domains restricted to so-called principal values).
6. Polynomial function: A function of the form $f(x)=a_{0}+a_{1}x+a_{2}x^{2}+\ldots + a_{n}x^{n}$, where n is zero or positive integer only and $a_{i} \in \Re$ is called a polynomial with real coefficients. Example. $f(x)=ax^{2}=bx+c$, where $a \neq 0$, $a, b, c \in \Re$ is called a quadratic function in x. (this is a general parabola).
7. Rational function: The function of the type $\frac{f(x)}{g(x)}$, where $g(x) \neq 0$, where $f(x)$ and $g(x)$ are polynomial functions of x, defined in a domain, is called a rational function. Such a function can have asymptotes, a term we define later. Example, $y=f(x)=\frac{1}{x}$, which is a hyperbola with asymptotes X and Y axes.
8. Absolute value function: Let $f: \Re \longrightarrow \Re$ be given by $f(x)=|x|=x$ when $x \geq 0$ and $f(x)=-x$, when $x<0$ for any $x \in \Re$. Note that $|x|=\sqrt{x^{2}}$ since the radical sign indicates positive root of a quantity by convention.
9. Signum function: Let $f: \Re \longrightarrow \Re$ where $f(x)=1$, when $x>0$ and $f(x)=0$ when $x=0$ and $f(x)=-1$ when $x<0$. Such a function is called the signum function. (If you can, discuss the continuity and differentiability of the signum function). Clearly, the domain of this function  is full $\Re$ whereas the range is $\{ -1,0,1\}$.
10. In part III of the blog series, we have already defined the floor function and the ceiling function. Further properties of these functions are summarized (and some with proofs in the following wikipedia links): (once again, if you can, discuss the continuity and differentiablity of the floor and ceiling functions): https://en.wikipedia.org/wiki/Floor_and_ceiling_functions
11. Exponential function: A function $f: \Re \longrightarrow \Re^{+}$ given by $f(x)=a^{x}$ where $a>0$ is called an exponential function. An exponential function is bijective and its inverse is the natural logarithmic function. (the logarithmic function is difficult to define, though; we will consider the details later). PS: Quiz: Which function has a faster growth rate — exponential or a power function ? Consider various parameters.
12. Logarithmic function: Let a be a positive real number with $a \neq 1$. If $a^{y}=x$, where $x \in \Re$, then y is called the logarithm of x with base a and we write it as $y=\ln{x}$. (By the way, the logarithmic function is used in the very much loved mp3 music :-))

Regards,

Nalin Pithwa

## Set Theory, Relations and Functions: Preliminaries: IV:

Problem Set based on previous three parts:

I) Solve the inequalities in the following exercises expressing the solution sets as intervals or unions of intervals. Also, graph each solution set on the real line:

a) $|x| <2$ (b) $|x| \leq 2$ (c) $|t-1| \leq 3$ (d) $|t+2|<1$ (e) $|3y-7|<4$(f) $|2y+5|<1$ (g) $|\frac{z}{5}-1| \leq 1$ (h) $| \frac{3}{2}z-1| \leq 2$ (i) $|3-\frac{1}{x}|<\frac{1}{2}$ (j) $|\frac{2}{x}-4|<3$ (k) $|2x| \geq 4$ (l) $|x+3| \geq \frac{1}{2}$ (m) $|1-x| >1$ (n) $|2-3x| > 5$ (o) $|\frac{x+1}{2}| \geq 1$ (p) $|\frac{3x}{5}-1|>\frac{2}{5}$

Solve the inequalities in the following exercises. Express the solution sets as intervals or unions of intervals and graph them. Use the result $\sqrt{a^{2}}=|a|$ as appropriate.

(a) $x^{2}<2$ (b) $4 \leq x^{2}$ (c) $4 (d) $\frac{1}{9} < x^{2} < \frac{1}{4}$ (e) $(x-1)^{2}<4$ (f) $(x+3)^{2}<2$ (g) $x^{2}-x<0$ (h) $x^{2}-x-2 \geq 0$

III) Theory and Examples:

i) Do not fall into the trap $|-a|=a$. For what real numbers a is the equation true? For what real numbers is it false?

ii) Solve the equation: $|x-1|=1-x$.

iii) A proof of the triangle inequality:

Give the reason justifying each of the marked steps in the following proof of the triangle inequality:

$|a+b|^{2}=(a+b)^{2}$…..why ?

$=a^{2}+2ab++b^{2}$

$\leq a^{2}+2|a||b|+b^{2}$….why ?

$\leq |a|^{2}+2|a||b|+|b|^{2}$….why?

$=(|a|+|b|)^{2}$….why ?

iv) Prove that $|ab|=|a||b|$ for any numbers a and b.

v) If $|x| \leq 3$ and $x>-\frac{1}{2}$, what can you say about x?

vi) Graph the inequality: $|x|+|y| \leq 1$

Questions related to functions:

I) Find the domain and range of each function:

a) $f(x)=1-\sqrt{x}$ (b) $F(t)=\frac{1}{1+\sqrt{t}}$ (c) $g(t)=\frac{1}{\sqrt{4-t^{2}}}$

II) Finding formulas for functions:

a) Express the area and perimeter of an equilateral triangle as a function of the triangle’ s side with length s.

b) Express the side length of a square as a function of the cube’s diagonal length d. Then, express the surface area  and volume of the cube as a function of the diagonal length.

c) A point P in the first quadrant lies on the graph of the function $f(x)=\sqrt{x}$. Express the coordinates of P as functions of the slope of the line joining P to the origin.

III) Functions and graphs:

Graph the functions in the questions below. What symmetries, if any, do the graphs have?

a) $y=-x^{3}$ (b) $y=-\frac{1}{x^{2}}$ (c) $y=-\frac{1}{x}$ (d) $y=\frac{1}{|x|}$ (e) $y = \sqrt{|x|}$ (f) $y=\sqrt{-x}$ (g) $y=\frac{x^{3}}{8}$ (h) $y=-4\sqrt{x}$ (i) $y=-x^{\frac{3}{2}}$ (j) $y=(-x)^{\frac{3}{2}}$ (k) $y=(-x)^{\frac{2}{3}}$ (l) $y=-x^{\frac{2}{3}}$

IV) Graph the following equations ad explain why they are not graphs of functions of x. (a) $|y|=x$ (b) $y^{2}=x^{2}$

V) Graph the following equations and explain why they are not graphs of functions of x: (a) $|x|+|y|=1$ (b) $|x+y|=1$

VI) Even and odd functions:

In the following questions, say whether the function is even, odd or neither.

a) $f(x)=3$ (b) $f(x=x^{-5}$ (c) $f(x)=x^{2}+1$ (d) $f(x)=x^{2}+x$ (e) $g(x)=x^{4}+3x^{2}-1$ (f) $g(x)=\frac{1}{x^{2}-1}$ (g) $g(x)=\frac{x}{x^{2}-1}$ (h) $h(t)=\frac{1}{t-1}$ (i) $h(t)=|t^{3}|$ (j) $h(t)=2t+1$ (k) $h(t)=2|t|+1$

Sums, Differences, Products and Quotients:

In the two questions below, find the domains and ranges of $f$, $g$, $f+g$, and $f-g$.

i) $f(x)=x$, $g(x)=\sqrt{x-1}$ (ii) $f(x)=\sqrt{x+1}$, $g(x)=\sqrt{x-1}$

In the two questions below, find the domains and ranges of $f$, $g$, $\frac{f}{g}$ and $\frac{g}{f}$

i) $f(x)=2$, $g(x)=x^{2}+1$

ii) $f(x)=1$, $g(x)=1+\sqrt{x}$

Composites of functions:

1. If $f(x)=x+5$, and $g(x)=x^{2}-5$, find the following: (a) $f(g(0))$ (b) $g(f(0))$ (c) $f(g(x))$ (d) $g(f(x))$ (e) $f(f(-5))$ (f) $g(g(2))$ (g) $f(f(x))$ (h) $g(g(x))$
2. If $f(x)=x-1$ and $g(x)=\frac{1}{x+1}$, find the following: (a) $f(g(\frac{1}{2}))$ (b) $g(f(\frac{1}{2}))$ (c) $f(g(x))$ (d) $g(f(x))$ (e) $f(f(2))$ (f) $g(g(2))$ (g) $f(f(x))$ (h) $g(g(x))$
3. If $u(x)=4x-5$, $v(x)=x^{2}$, and $f(x)=\frac{1}{x}$, find formulas or formulae for the following: (a) $u(v(f(x)))$ (b) $u(f(v(x)))$ (c) $v(u(f(x)))$ (d) $v(f(u(x)))$ (e) $f(u(v(x)))$ (f) $f(v(u(x)))$
4. If $f(x)=\sqrt{x}$, $g(x)=\frac{x}{4}$, and $h(x)=4x-8$, find formulas or formulae for the following: (a) $h(g(f(x)))$ (b) $h(f(g(x)))$ (c) $g(h(f(x)))$ (d) $g(f(h(x)))$ (e) $f(g(h(x)))$ (f) $f(h(g(x)))$

Let $f(x)=x-5$, $g(x)=\sqrt{x}$, $h(x)=x^{3}$, and $f(x)=2x$. Express each of the functions in the questions below as a composite involving one or more of f, g, h and j:

a) $y=\sqrt{x}-3$ (b) $y=2\sqrt{x}$ (c) $y=x^{\frac{1}{4}}$ (d) $y=4x$ (e) $y=\sqrt{(x-3)^{3}}$ (f) $y=(2x-6)^{3}$ (g) $y=2x-3$ (h) $y=x^{\frac{3}{2}}$ (i) $y=x^{9}$ (k) $y=x-6$ (l) $y=2\sqrt{x-3}$ (m) $\sqrt{x^{3}-3}$

Questions:

a) $g(x)=x-7$, $f(x)=\sqrt{x}$, find $(f \circ g)(x)$

b) $g(x)=x+2$, $f(x)=3x$, find $(f \circ g)(x)$

c) $f(x)=\sqrt{x-5}$, $(f \circ g)(x)=\sqrt{x^{2}-5}$, find $g(x)$.

d) $f(x)=\frac{x}{x-1}$, $g(x)=\frac{x}{x-1}$, find $(f \circ g)(x)$

e) $f(x)=1+\frac{1}{x}$, $(f \circ g)(x)=x$, find $g(x)$.

f) $g(x)=\frac{1}{x}$, $(f \circ g)(x)=x$, find $f(x)$.

Reference: Calculus and Analytic Geometry, G B Thomas.

NB: I have used an old edition (printed version) to prepare the above. The latest edition may be found at Amazon India link:

https://www.amazon.in/Thomas-Calculus-George-B/dp/9353060419/ref=sr_1_1?crid=1XDE2XDSY5LCP&keywords=gb+thomas+calculus&qid=1570492794&s=books&sprefix=G+B+Th%2Caps%2C255&sr=1-1

Regards,

Nalin Pithwa

## Ceiling and floor functions: IITJEE mains training

Problem 1:

For what values of x, is (a) $\lfloor x \rfloor =0$ (b) $\lceil x \rceil =0$?

Problem 2:

Which real numbers x satisfy the equation $\lfloor x \rfloor = \lceil x \rceil$?

Problem 3:

Does $\lceil (-x) \rceil = - (\lfloor x \rfloor)$ for all real x? Give reasons for your answer.

Problem 4:

Graph: $f(x)=\lfloor x \rfloor$ when $x \geq 0$; and $f(x) = \lceil x \rceil$, when $x <0$.

Why is f(x) called the integer part of x? Discuss the continuity and differentiability of f(x).

Cheers,

Nalin Pithwa

## Various proofs of important algebraic identity: a^{3}+b^{3}+c^{3}-3abc

We know the following factorization: $a^{3}+b^{3}+c^{3}-3abc= (a+b+c)(a^{2}+b^{2}+c^{2}-ab-bc-ca)$

Proof 1:

Let a, b, c be roots of a polynomial P(X) Then by fundamental theorem of algebra

$P(X)=(X-a)(X-b)(X-c) = X^{3}-(a+b+c)X^{2}+(ab+bc+ca)X-abc$.

Now, once again basic algebra says that as a, b, c are roots/solutions of the above:

$P(a)=a^{3}-(a+b+c)a^{2}+(ab+bc+ca)a-abc=0$

$P(b)=b^{3}-(a+b+c)b^{2}+(ab+bc+ca)b-abc=0$

$P(c)=c^{3}-(a+b+c)c^{2}+(ab+bc+ca)c-abc$=0\$

$0= a^{3}+b^{3}+c^{3}-(a+b+c)(a^{2}+b^{2}+c^{2})+(ab+bc+ca)(a+b+c)-3abc$

So, we get $a^{3}+b^{3}+c^{3}-3abc=(a+b+c)(a^{2}+b^{2}+c^{2}-ab-bc-ca)$

Also, the above formula can be written as $a^{3}+b^{3}+c^{3}-3abc=(a+b+c)(\frac{1}{2})((a-b)^{2}+(b-c)^{2}+(c-a)^{2})$

Proof 2:

Consider the following determinant D: $\left| \begin{array}{ccc} a & b & c\\c & a & b\\ b & c & a \end{array} \right|$

On adding all three columns to the first column: we know that the value of the determinant is unchanged: So we get the following:

$D = \left| \begin{array}{ccc} a+b+c & b & c \\a+b+c & a & b\\a+b+c & c & a \end{array} \right|$. Note that columns 2 and 3 of the three by three determinant do not change.

On expanding the original determinant D, we get

$D = a(a^{2}-bc)-b(ac-b^{2})+c(c^{2}-ab)$

$D= a^{3}-abc-bac+b^{3}+c^{3}-cab$

$D= a^{3}+b^{3}+c^{3}-3abc$

Whereas we get from the other transformed but equal D:

$D =(a+b+c) \left| \begin{array}{ccc} 1 & b & c \\ 1 & a & b \\ 1 & c & a \end{array}\right|$

$D=(a+b+c)((a^{2}-bc)-b(a-b)+c(c-a))$

So, that we again get $a^{3}+b^{3}+c^{3}-3abc = (a+b+c)(a^{2}+b^{2}+c^{2}-ab-bc-ca)$

Proof 3:

Now, let us consider $E=a^{2}+b^{2}+c^{2}-ab-bc-ca$ as a quadratic in a with b and c as parameters.

That is, $E= a^{2} -(b+c)a + b^{2}+c^{2}-bc$

Then, the discriminant is given by

$\triangle = (b+c)^{2}-4 \times 1 \times (b^{2}+c^{2}-bc)$, which in turn equals:

$\triangle = (b^{2}+c^{2}+2bc)-4(b^{2}+c^{2}-bc) = -3b^{2}-3c^{2}+6bc=-3(b-c)^{2}$

$a_{1}, a_{2} = \frac{(b+c) \pm i\sqrt{3}(b-c)}{2}$

$a_{1}= b(\frac{1+i\sqrt{3}}{2})=c(\frac{1-i\sqrt{3}}{2}) = -b\omega - c\omega^{2}$

$a_{2}=b(\frac{1-i\sqrt{3}}{2})+c(\frac{1-i\sqrt{3}}{2})=-b\omega^{2}-c\omega$

Hence, the factorization of the above quadratic in a is given as:

$a^{2}+b^{2}+c^{2}-ab-bc-ca = (a+b\omega+c\omega^{2})(a+b\omega^{2}+c\omega)$

So, the other non-trivial factorization of the above famous algebraic identity is:

$a^{3}+b^{3}+c^{3}-3abc=(a+b+c)(a+b\omega+ c\omega^{2})(a+b\omega^{2}+c\omega)$

where $1, \omega, \omega^{2}$ are cube roots of unity.

Proof 4:

Factorize the expression $a^{3}+b^{3}+c^{3}-3abc by (a+b+c)$.

Solution 4:

We can carry out the above polynomial division by considering the dividend to be a polynomial in a single variable, say a, (and assuming b and c are just parameters; so visualize them as arbitrary but fixed constants); further arrange the dividend in descending powers of a; so also arrange the divisor in descending powers of a (well, of course, it is just linear in a; and assume b and c are parameters also in dividend).

Proof 5:

Prove that the eliminant of

$ax+cy+bz=0$

$cx+by+az=0$

$bx+ay+cz=0$

is $a^{3}+b^{3}+c^{3}-3abc=0$

Proof 5:

By Cramer’s rule, the eliminant is given by determinant $\left| \begin{array}{ccc}a & c & b \\ c & b & a\\b & a & c \end{array}\right|=0$.

On expansion using the first row:

$a(bc-a^{2})-c(c^{2}-ab)+b(ac-b^{2})=0$

$a^{3}+b^{3}+c^{3}-3abc=0$ upon multiplying both the sides of the above equation by (-1). Of course, we have only been able to generate the basic algebraic expression but we have done so by encountering a system of three linear equations in x, y, z. (we could append any of the above factorization methods to this further!! :-)))

Cheers,

Nalin Pithwa