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}

II) Quadratic Inequalities:

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<x^{2}<9 (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$

Adding all the above:

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

The personality of Leonhard Euler

The portrait of Euler that emerges from his publications and letters is that of a genial man of simple tastes and conventional religious faith. He was even wealthy, at least in the second half of his life, but ostentation was not part of his lifestyle. His memory was prodigious, and contemporary accounts have emphasized this. He would delight relatives, friends, and acquaintances with a literal recitation of any song from Virgil’s Aenesis, and he would remember minutes of Academy meetings years after they were held. He was not given to envy, and when someone made an advance on his work his happiness was genuine. For example, when he learnt of Lagrange’s improvements on his work on elliptic integrals, he wrote to him that his admiration knew no bounds and then proceeded to improve upon Lagrange!

But, what is most characteristic of his work is its clarity and openness. He never tries to hide the difficulties from the reader. This is in stark contrast to Newton, who was prone to hide his methods in obscure anagrams, and even from his successor, Gauss, who very often erased his steps to present a monolithic proof that was seldom illuminating. In Euler’s writings there are no comments on how profound his results are, and in his papers one can follow his ideas step by step with the greatest of ease. Nor was he chary of giving credit to others; his willingness to share his summation formula with Maclaurin, his proper citations to Fuguano when he started his work on algebraic integrals, his open admiration for Lagrange when the latter improved on his work in calculus of variations are all instances of his serene outlook. One can only contrast this with Gauss’s reaction to Bolyai’s discovery of non-Euclidean postulates. Euler was secure in his knowledge of what he had achieved but never insisted that he should be the only one on top of the mountain.

Perhaps, the most astounding aspect of his scientific opus is its universality. He worked on everything that had any bearing on mathematics. For instance, his early training under Johann Bernoulli did not include number theory; nevertheless, within a couple of years after reaching St. Petersburg he was deeply immersed in it, recreating the entire corpus of Fermat’s work in that area and then moving well beyond him. His founding of graph theory as a separate discipline, his excursions in what we call combinatorial topology, his intuition that suggested to him the idea of exploring multizeta values are all examples of a mind that did not have any artificial boundaries. He had no preferences about which branch of mathematics was dear to him. To him, they were all filled with splendour, or Herrlichkeit, to use his own favourite word.

Hilbert and Poincare were perhaps last of the universalists of modern era. Already von Neumann had remarked that it would be difficult even to have a general understanding of more than a third of the mathematicians of his time. With the explosive growth of mathematics in the twentieth century we may never see again the great universalists. But who is to say what is and is not possible for the human mind?

It is impossible to read Euler and not fall under his spell. He is to mathematics what Shakespeare is to literature and Mozart to music: universal and sui generis.

Reference:

Euler Through Time: A New Look at Old Themes by V S Varadarajan:

Hindustan Book Agency;

http://www.hindbook.com/index.php/euler-through-time-a-new-look-at-old-themes;

Amazon India link:

https://www.amazon.in/Euler-Through-Time-Look-Themes/dp/9380250592/ref=sr_1_1?keywords=Euler+Through+Time&qid=1568316624&s=books&sr=1-1

 

 

Rules for Inequalities

If a, b and c are real numbers, then

  1. a < b \Longrightarrow a + c< b + c
  2. a < b \Longrightarrow a - c < b - c
  3. a < b \hspace{0.1in} and \hspace{0.1in}c > 0 \Longrightarrow ac < bc
  4. a < b \hspace{0.1in} and \hspace{0.1in}c < 0 \Longrightarrow bc < ac special case: a < b \Longrightarrow -b < -a
  5. a > 0 \Longrightarrow \frac{1}{a} > 0
  6. If a and b are both positive or both negative, then a < b \Longrightarrow \frac{1}{b} < \frac{1}{a}.

Remarks:

Notice the rules for multiplying an inequality by a number: Multiplying by a positive number preserves the inequality; multiplying by a negative number reverses the inequality. Also, reciprocation reverses the inequality for numbers of the same sign.

Regards,

Nalin Pithwa.

Set Theory, Relations, Functions Preliminaries: Part III

FUNCTIONS:

Functions as a special kind of relation:

Let us first consider an example where set A is a set of men, and B is a set of positive real numbers. Let us say f is a relation from A to B given by : f = \{ (x,y) : x \in A, y \hspace{0.1in} is \hspace{0.1in} the \hspace{0.1in} weight \hspace{0.1in} of \hspace{0.1in} the \hspace{0.1in} person \hspace{0.1in} x \}

Hence, f “relates” every man in set A to his weight in set B. That is,

i) Every man has some weight associated with him in set B. (ii) That weight is unique. That is, a person cannot have more than one weight (at a given time, of course) !! 🙂 This, of course, does not mean that two different persons, say P and Q may not have the same weight 100 kg ( the same element of set B). The only thing it means is that any one person, say P will have one and only one weight (100kg) at the time instant of measurement and not more than one weights (which would be crazy) at a time instant it is measured !!

Definition I (a function defined as a relation):

A function f from a set A (called domain) to a set B (called codomain) is a relation that associates or “pairs up” every element of domain A with a unique element of codomain B. (Note that whereas a relation from a set A to a set B is just a subset of the cartesian product A \times B).

Some remarks: The above definition is also motivated by an example of a function as a relation. On the other hand, another definition of a function can be motivated as follows:

We know that the boiling point of water depends on the height of water above sea level. We also know that the simple interest on a deposit in a bank depends on the duration of deposit held in the bank. In these and several such examples, one quantity, say y, depends on another quantity “x”.

Symbol: f: A \longrightarrow B; if x \in A, y \in B, then we also denote: f: x \longmapsto y; we also write y=f(x), read as “y is f of x”.

Here, y is called image of x under f and x is called the preimage of y under f.

Definition: Range: The set of all images in B is called the range of f. That is, Range = \{ f(x): x \in A\}

Note: (i) Every function is a relation but every relation need not be a function. (Homework quiz: find illustrative examples for the same) (ii) If the domain and codomain are not specified, they are assumed to be the set of real numbers.

In calculus, we often want to refer to a generic function without having any particular formula in mind. Leonhard Euler invented a symbolic way to say “y is a function of x” by writing

y = f(x) (“y equals f of x”)

In this equation, the symbol f represents the function. The letter x, called the independent variable, represents an input value from the domain of f, and y, the dependent variable, represents the corresponding output value f(x) in the range of f. Here is the formal definition of function: (definition 2):

function from a set D to a set \Re is a rule that assigns a unique element f(x) in \Re to each element x in D.

In this definition, D=D(f) (read “D of f”) is the domain of the function f and \Re is the range (or codomain containing the range of f).

Think of a function f as a kind of machine that produces an output value f(x) in its range whenever we feed it an input value x from its domain. In our scope, we will usually define functions in one of two ways:

a) by giving a formula such as y=x^{2} that uses a dependent variable y to denote the value of the function, or

b) by giving a formula such as f(x)=x^{2} that defines a function symbol f to name the function.

NOTE: there can be well-defined functions which do not have any formula at all; for example, let f(x) = 0 when x \in Q and f(x)=1, when x \in Q^{'}.

Strictly speaking, we should call the function f and not f(x) as the latter denotes the value of the function at the point x. However, as is common usage, we will often refer to the function as f(x) in order to name the variable on which f depends.

It is sometimes convenient to use a single letter to denote both a function and the dependent variable. For instance, we might say that the area A of a circle of radius r is given by the function : A(r)=\pi r^{2}.

Evaluation:

As we said earlier, most of the functions in our scope will be real-valued function of a real variable, functions whose domains and ranges are sets of real numbers. We evaluate such functions by susbtituting particular values from the domain into the function’s defining rule to calculate the corresponding values in the range.

Example 1:

The volume V of a ball (solid sphere) r is given by the function: V(r)=\frac{4}{3}\pi  r^{3}.

The volume of a ball of radius 3 meters is : V(3)=\frac{4}{3}\pi (3)^{3}=36 \pi m^{3}.

Example 2:

Suppose that the function F is defined for all real numbers t by the formula: F(t)=2(t-1)+3.

Evaluate F at the output values 0, 2, x+2, and F(2).

Solution 2:

In each case, we substitute the given input value for t into the formula for F:

F(0)=2(0-1)+3=-2+3=1

F(2)=2(2-1)+3=2+3=5

F(x+2)=2(x+2-1)+3=2x+3

F(F(2))=F(5)=2(5-1)+3=11

The Domain Convention

When we define a function y=f(x) with a formula and the domain is not stated explicitly, the domain is assumed to be the largest set of x-values for which the formula gives real x-values. This is the function’s so-called natural domain. If we want the domain to be restricted in some way, we must say so.

The domain of the function y=x^{2} is understood to be the entire set of real numbers. The formula gives a real value y-value for every real number x. If we want to restrict the domain to values of x greater than or equal to 2, we must write ” y=x^{2}” for x \geq 2.

Changing the domain to which we apply a formula usually changes the range as well. The range of y=x^{2} is [0, \infty). The  range of y=x^{2} where x \geq 2 is the set of all numbers obtained by squaring numbers greater than or equal to 2. In symbols, the range is \{ x^{2}: x \geq 2\} or \{ y: y \geq 4\} or [4,\infty)

Example 3:

Function : y = \sqrt{1-x^{2}}; domain [-1,1]; Range (y) is [0,1]

Function: y=\frac{1}{x}; domain (-\infty,0) \bigcup (0,\infty); Range (y) is (-\infty,0)\bigcup (0,\infty)

Function: y=\sqrt{x}; domain (0,\infty) and range (y) is (0,\infty)

Function y = \sqrt{4-x}, domain (-\infty,,4], and range (y) is [0, \infty)

Graphs of functions:

The graph of a function f is the graph of the equation y=f(x). It consists of the points in the Cartesian plane whose co-ordinates (x,y) are input-output pairs for f.

Not every curve you draw is the graph of a function. A function f can have only one value f(x) for each x in its domain so no vertical line can intersect the graph of a function more than once. Thus, a circle cannot be the graph of a function since some vertical line intersect the circle twice. If a is in the domain of a function f, then the vertical line x=a will intersect the graph of f in the single point (a, f(a)).

Example 4: Graph the function y=x^{2} over the interval [-2.2]. (homework).Thinking further: so plotting the above graph requires a table of x and y values; but how do we connect the points ? Should we connect two points by a straight line, smooth line, zig-zag line ??? How do we know for sure what the graph looks like between the points we plot? The answer lies in calculus, as we will see in later chapter. There will be a marvelous mathematical tool called the derivative to find a curve’s shape between plotted points. Meanwhile, we will have to settle for plotting points and connecting them as best as we can. 

PS: (1) you can use GeoGebra, a beautiful freeware for plotting various graphs, and more stuff https://www.geogebra.org/ (2) If you wish, you can use a TI-graphing calculator. This is a nice investment for many other things like number theory also. See for example,

https://www.amazon.in/Texas-Instruments-Nspire-Graphing-Calculator/dp/B004NBZAYS/ref=sr_1_2?crid=3JSHJUOZMDMUS&keywords=ti+nspire+cx&qid=1569334614&s=electronics&sprefix=TI+%2Caps%2C267&sr=1-2

Meanwhile, you need to be extremely familiar with graphs of following functions; plot and check on your own:

y=x^{3}, y=x^{2/3}, y=\sqrt{x}, y=\sqrt[3]{x}, y=\frac{1}{x}, y=\frac{1}{x^{2}}, y=mx, where m \in Z, y=x^{3/2}

Sums, Differences, Products and Quotients

Like numbers, functions can be added, subtracted, multiplied and divided (except where the the denominator is zero) to produce new functions. If f and g are functions, then for every x that belongs to the domains of BOTH f and g, we define functions: f+g, f-g, fg by the formulas:

(f+g)(x)=f(x)+g(x),

(f-g)(x)=f(x)-g(x)

(fg)(x)=f(x)g(x)

At any point D(f) \bigcap D(g) at which g(x) \neq 0, we can also define the function f/g by the formula:

(\frac{f}{g})(x)=\frac{f(x)}{g(x)}, where g(x) \neq 0

Functions can also be multiplied by constants. If c is a real number, then the function cf is defined for all x in the domain of f by (cf)(x)=cf(x)

Example 5:

Function f, formula y=\sqrt{x}, domain [0,\infty)

Function g, formula g(x)=\sqrt{(1-x)}, domain (-\infty, 1]

Function 3g, formula 3g(x)=3\sqrt{(1-x)}, domain (-\infty, 1]

Function f+g, formula (f+g)(x)=\sqrt{x}+\sqrt{(1-x)}, domain [0,1]=D(f) \bigcap D(g)

Function f-g, formula (f-g)(x)=\sqrt{x}-\sqrt{(1-x)}, domain [0.1]

Function g-f, formula (g-f)(x)=\sqrt{(1-x)}-\sqrt{x}, domain [0,1]

Function f . g, formula (f . g)(x)=f(x)g(x) = \sqrt{x(1-x)}, domain [0,1]

Function \frac{f}{g}, formula \frac{f}{g}(x)=\frac{f(x)}{g(x)}=\sqrt{\frac{x}{1-x}}, domain is [0,1)

Function \frac{g}{f}(x) = \frac{g(x)}{f(x)}=\sqrt{\frac{1-x}{x}}, domain (0,1]

Composite Functions:

Composition is another method for combining functions.

Definition:

If f and g are functions, the composite function f \circ g (f “circle” g) is defined by (f \circ g)(x)=f(g(x)). The domain of f \circ g consists of the numbers x in the domain of g for which g(x) lies in the domain of f.

The definition says that two functions can be composed when the image of the first lies in the domain of the second. To (f \circ g)(x) we first find g(x) and second find f(g(x)).

Clearly, in general, (f \circ g)(x) \neq (g \circ f)(x). That is, composition of functions is not commutative.

Example 6:

If f(x)=\sqrt{x} and g(x)=x+1, find (a) (f \circ g)(x) (b) (g \circ f)(x) (c) (f \circ f)(x) (d) (g \circ g)(x)

Solution 6:

a) (f \circ g)(x) = f(g(x))=\sqrt{g(x)}=\sqrt{x+1}, domain is [-1, \infty)

b) (g \circ f)(x)=g(f(x))=f(x)+1=\sqrt{x}+1, domain is [0, \infty)

c) (f \circ f)(x)=f(f(x))=\sqrt{f(x)}=\sqrt{\sqrt{x}}=x^{\frac{1}{4}}, domain is [0, \infty)

d) (g \circ g)(x)=g(g(x))=g(x)+1=(x+1)+1=x+2, domain is \Re or (-\infty, \infty)

Even functions and odd functions:

A function f(x) is said to be even if f(x)=f(-x). That is, the function possesses symmetry about the y-axis. Example, y=f(x)=x^{2}.

A function f(x) is said to be odd if f(x)=-f(-x). That is, the function possesses symmetry about the origin. Example y=f(x)=x^{3}.

Any function can be expressed as a sum of an even function and an odd function.

A function could be neither even nor odd.

Note that a function like y^{2}=x possesses symmetry about the x-axis !!

Piecewise Defined Functions:

Sometimes a function uses different formulas or formulae over different parts of its domain. One such example is the absolute value function:

y=f(x) = |x|=x, when x \geq 0 and y=-x, when x<0.

Example 7:

The function f(x)=-x, when x<0, y=f(x)=x^{2}, when 0 \leq x \leq 1, and f(x)=1, when x>1.

Example 8:

The greatest integer function:

The function whose value at any number x is the greatest integer less than or equal to x is called the greatest integer function or the integer floor function. It is denoted by \lfloor x \rfloor.

Observe that \lfloor 2.4 \rfloor =2; \lfloor 1.4 \rfloor =1; \lfloor 0 \rfloor =0; \lfloor -1.2 \rfloor =-2; \lfloor 2 \rfloor =2; \lfloor 0.2 \rfloor =0\lfloor -0.3 \rfloor =-1; \lfloor -2 \rfloor =-2.

Example 9:

The least integer function:

The function whose value at any number x is the smallest integer greater than or equal to x is called the least integer function or the integer ceiling function. It is denoted by \lceil x \rceil. For positive values of x, this function might represent, for example, the cost of parking x hours in a parking lot which charges USD 1 for each hour or part of an hour.

Cheers,

Nalin Pithwa