## Category Archives: IITJEE Foundation mathematics

### Skill Check II: IITJEE foundation maths

Set Theory Primer/basics/fundamentals/preliminaries:

I. Represent the following sets in Venn Diagrams: (a) $\Xi = \{ x : x = n, n <40, n \in \mathcal{N} \}$ (b) $\mu = \{ x : x = 6n, n < 6. n \in \mathcal{N}\}$ (c) $\alpha = \{ x: x = 3n, n<8, n \in \mathcal{N} \}$

2. If $x = \{ x: x<29 \hspace{0.1in}and \hspace{0.1in}prime\}$ and $A = \{ x: x \hspace{0.1in} is \hspace{0.1in} a \hspace{0.1in} prime \hspace{0.1in} factor \hspace{0.1in} of \hspace{0.1in} 210\}$, represent A in Venn diagram and find $\overline{A}$.

3. 96 boys of a school appeared for a physical for selection in NCC and Boy Scouts. 21 boys got selected in both NCC and Boy Scouts, 44 boys were not selected in Boy Scouts and 20 boys were not selected only in boy scouts. Draw Venn diagram and find : (i) how many boys did not get selected in NCC and boy scouts. (ii) how many boys did not get selected only in NCC (iii) how many boys got selected in NCC (iv) how many boys got selected in boy Scouts (v) How many boys got selected in NCC not in boy Scouts?

Regards,

Nalin Pithwa.

### IITJEE Foundation Maths: Tutorial Problems IV

1. Resolve into factors: (a) $2x^{2}-3ab+(a-6b)x$ (b) $4x^{2}-4xy-15y^{2}$
2. In the expression, $x^{3}-2x^{2}+3x-4$, substitute $a-2$ for x, and arrange the result according to the descending powers of a.
3. Simplify: (i) $\frac{x}{1-\frac{1}{1+x}}$ (ii) $\frac{x^{2}}{a+\frac{x^{2}}{a+\frac{x^{2}}{a}}}$
4. Find the HCF of $3x^{3}-11x^{2}+x+15$ and $5x^{4}-7x^{3}-20x^{2}-11x-3$
5. Express in the simplest form: (i) $\frac{\frac{x}{y}-\frac{y}{x}}{\frac{x+y}{y}-\frac{y+x}{x}}$ (ii) $(\frac{x^{3}-1}{x-1} + \frac{x^{3}+1}{x+1})\div (\frac{1}{x-1} + \frac{1}{x+1})$
6. A person possesses Rs. 5000 stock, some at 3 per cent; four times as much at 3.5 %, and the rest at 4 %; find the amount of each kind of stock when his income is Rs. 178.
7. Simplify the expression: $-3[(a+b)-{(2a-3b) -( 5a+7b-16c) - (-13a +2b -3c -5d)}]$, and find its value when $a=1, b=2, c=3, d=4$.
8. Solve the following equations : (i) $11y-x=10$ and $11x-101y=110$ (ii) $x+y-z=3$, and $x+z-y=5$, and $y+z-x=7$.
9. Express the following fractions in their simplest form: (i) $\frac{32x^{3}-2x+12}{12x^{5}-x^{4}+4x^{2}}$ (ii) $\frac{1}{x + \frac{1}{1+ \frac{x+3}{2-x}}}$
10. What value of a will make the product of $3-8a$ and $3a+4$ equal to the product of $6a+11$ and $3-4a$?
11. Find the LCM of $x^{3}-x^{2}-3x-9$ and $x^{3}-2x^{2}-5x-12$
12. A certain number of two digits is equal to seven times the sum of its digits; if the digit in the units’ place be decreased by two and that in the tens place by one, and if the number thus formed be divided by the sum of its digits, the quotient is 10. Find the number.
13. Find the value of $\frac{6x^{2}-5xy-6y^{2}}{2x^{2}+xy-y^{2}} \times \frac{3x^{2}-xy-4y^{2}}{2x^{2}-5xy+3y^{2}} \div \frac{9x^{2}-6xy-8y^{2}}{2x^{2}-3xy+y^{2}}$
14. Resolve each of the following expressions into four factors: (i) $4a^{4}-17a^{2}b^{2}+4b^{4}$; (ii) $x^{8}-256y^{8}$
15. Find the expression of highest dimensions which will divide $24a^{4}b -2a^{2}b^{2}-9ab^{4}$ and $18a^{6}+a^{4}b^{2}-6a^{3}b^{3}$ without remainder.
16. Find the square root of : (i) $x(x+1)(x+2)(x+3)+1$ (ii) $(2a^{2}+13a+15)(a^{2}+4a-5)(2a^{2}+a-3)$
17. Simplify: $x - \frac{2x-6}{x^{2}-6x+9} - 3 + \frac{x^{2}+3x-4}{x^{2}=x-12}$
18. A quantity of land, partly pasture and partly arable, is sold at the rate of Rs. 60 per acre for the pasture and Rs. 40 per acre for the arable, and the whole sum obtained at Rs. 10000. If the average price per acre were Rs. 50, the sum obtained would be 10 per cent higher; find how much of the land is pasture and how much is arable.

Regards,

Nalin Pithwa

Purva building, 5A
Flat 06
Mumbai, Maharastra 400101
India

### IITJEE Foundation Maths : Tutorial Problems III

1. When $a=-3, b=5, c=-1, d=0$, find the value of $26c\sqrt[3]{a^{3}-c^{2}d+5bc-4ac+d^{2}}$
2. Solve the equations: (a) $\frac{1}{3}x - \frac{2}{7}y = 8 -2x$ and $\frac{1}{2}y - 3x =3-y$ (b) $1 = y+z =2(z+x)=3(x+y)$
3. Simplify: (a) $\frac{a-x}{a+x} - \frac{4x^{2}}{a^{2}-x^{2}} + \frac{a-3x}{x-a}$ (b) $\frac{b^{2}-3b}{b^{2}-2b+4} \times \frac{b^{2}+b-30}{b^{2}+3b-18} \div \frac{b^{3}-3b^{2}-10b}{b^{2}+8}$
4. Find the square root of : $9-36x+60x^{2}-\frac{160}{3}x^{3}+\frac{80}{3}x^{4}-\frac{64}{3}x^{5}+\frac{64}{81}x^{6}$
5. In a cricket match, the extras in the first innings are one-sixteenth of the score, and in the second innings the extras are one-twelfth of the score. The grand total is 296, of which 21 are extras; find the score in each innings.
6. Find the value of $\frac{a^{2}-x^{2}}{\frac{1}{a^{2}} - \frac{2}{ax} + \frac{1}{x^{2}}} \times \frac{\frac{1}{a^{2}x^{2}}}{a+x}$
7. Find the value of : $\frac{1}{3}(a+2) -3(1-\frac{1}{6}b) - \frac{2}{3}(2a-3b+\frac{3}{2}) + \frac{3}{2}b - 4(\frac{1}{2}a-\frac{1}{3})$.
8. Resolve into factors: (i) $3a^{2} -20a-7$ (ii) $a^{4}b^{2}-b^{4}a^{2}$
9. Reduce to lowest terms: $\frac{4x^{3}+7x^{2}-x+2}{4x^{3}+5x^{2}-7x-2}$
10. Solve the following equations: (a) $x-6 -\frac{x-12}{3}= \frac{x-4}{2} + \frac{x-8}{4}$; (b) $x+y-z=0$, $x-y+z=4$, $5x+y+z=20$; (c) $\frac{ax+b}{c} + \frac{dx+e}{f} =1$
11. Simplify: $\frac{x+3}{x^{2}-5x+6} - \frac{x+2}{x^{2}-9x+14} + \frac{4}{x^{2}-10x+21}$
12. A purse of rupees is divided amongst three persons, the first receiving half of them and one more, the second half of the remainder and one more, and the third six. Find the number of rupees the purse contained.
13. If $h=-2, k=1, l=0, m=1, n=-3$, find the value of $\frac{h^{2}(m-l)-\sqrt{3hn}+hk}{m(l-h)-2hm^{2}+\sqrt[3]{4hk}}$
14. Find the LCM of $15(p^{3}+q^{3}), 5(p^{2}-pq+q^{2}), 4(p^{2}+pq+q^{2}), 6(p^{2}-q^{2})$
15. Find the square root of (i) $\frac{4x^{2}}{9} + \frac{9}{4x^{2}} -2$; (ii) $1-6a+5a^{2}+12a^{3}+4a^{4}$
16. Simplify $\frac{20x^{2}+27x+9}{15x^{2}+19x+6} + \frac{20x^{2}+27x+9}{12x^{2}+17x+6}$
17. Solve the equations: (i) $\frac{a(x-b)}{a-b} + \frac{b(x-a)}{b-a} =1$ (ii) $\frac{9}{x-4} + \frac{3}{x-8} = \frac{4}{x-9} + \frac{8}{x-3}$
18. A sum of money is to be divided among a number of persons; if Rs. 8 is given to each there will be Rs. 3 short, and if Rs. 7.50 is given to each there will be Rs. 2 over; find the number of persons.

Regards,

Nalin Pithwa

Purva building, 5A
Flat 06
Mumbai , Maharastra 400101
India

### IITJEE Foundation Maths: Tutorial Problems II

1. A, B, C start from the same place at the rates a, $a+b$, $a+2b$ kilometres per hour respectively. B starts n hours after A, how long after B must C start in order that they may overtake A at the same instant, and how far will they then have walked?
2. Find the distance between two towns when by increasing the speed 7 kilometres per hour a train can perform the journey in 1 hour less, and by reducing the speed 5 kilometres per hour can perform the journey in 1 hour more.
3. A person buys a certain quantity of land. If he had bought 7 hectares more each hectare would have cost Rs 80 less; and if each hectare had cost Rs. 360 more, he would have obtained 15 hectares less, how much did he pay for the land?
4. A can walk half a kilometre per hour faster than B; and three quarters of a kilometre per hour faster than C. To walk a certain distance C takes three-quarters of an hour more than B, and two hours more than A; find their rates of walking per hour.
5. A person spends Rs. 15 in buying goods; if each kg had cost 25 paise more he would have got 5 kg less, but if each kg had cost 15 paise less, he would have received 5 kg more; what weight did he buy?
6. Five silver coins weight 125 gm and are worth Rs. 6. Ten bronze coins weigh 500 gm and are worth 80 paise. A number of silver and bronze coins which are worth Rs. 134 weigh 11 kg and 250 gm. How many coins of each kind are there?
7. A and B are playing for money; in the first game, A loses one half of his money, but in the second he wins one-quarter of what B then has. When they cease playing, A has won Rs. 6 and B has still Rs. 14.50 more than A; with what amounts did they begin?
8. A, B, C each spend the same amount in buying different qualities of the same commodity. B pays 36 paise per kg less than A and obtains 750 gm more; C pays 60 paise per kg more than A and obtains one kg less; how much does each spend?

### IITJEE Foundation practice or training problem sheet: I

1. If the numerator of a fraction is increased by 5, it reduces to $\frac{2}{3}$, and if the denominator is increased by 9, it reduces to $\frac{1}{3}$. Find the fraction.
2. Find a fraction such that it reduces to $\frac{3}{5}$ if 7 is subtracted from its denominator, and reduces to $\frac{3}{8}$ on subtracting 3 from its numerator.
3. If unity is taken from the denominator of a fraction, it reduces to $\frac{1}{2}$; if 3 is added to the numerator it reduces to $\frac{4}{7}$, find the required fraction.
4. Find a fraction which becomes $\frac{3}{4}$ on adding 5 to the numerator and subtracting 1 from its denominator; and, reduces to $\frac{1}{3}$ on subtracting 4 from the numerator and adding 7 to the denominator.
5. If 9 is added to the numerator a certain fraction will be increased by $\frac{1}{3}$; if 6 is taken from the denominator the fraction reduces to $\frac{2}{3}$; find the required fraction.
6. At what time between 9 and 10 o’clock are the hands of a watch together?
7. When are the hands of a clock 8 minutes apart between the hours at 5 and 6 ?
8. At what time between 10 and 11 o’clock is the hour hand six minutes ahead of the minute hand?
9. At what time between 1 and 2 o’clock are the hands of a watch in the same straight line?
10. At what times between 12 and 1 o’clock are the hands of a watch at right angles?
11. A person buys 20 m of cloth and 25 m of canvas for Rs. 22.50. By selling the cloth at a gain of 15 per cent, and the canvas at a gain of 20 per cent, he clears Rs. 3.75. Find the price of each per metre.
12. A dealer spends Rs. 6950 in buying horses at Rs, 250/- each and cows at Rs. 200/- each; through disease, he loses 20 percent of the horses and 25 % of the cows. By selling the animals at the price he gave for them, he receives Rs. 5400/-. Find how many of each kind he bought.
13. The population of a certain district is 53000, of whom 835 can neither read nor write. These consists of 2 %, of all the males and 3 % of all the females; find the number of males and females.
14. Two persons C and D start simultaneously from two places a kilometre apart, and walk to meet each other; if C walks p kilometres per hour, and D one kilometre per hour faster than C, how far will D have walked when they meet?
15. A can walk a kilometres per hour faster than B; supposing that he gives B a start of c kilometres, and that B walks a kilometres per hour, how far will A have walked when he overtakes B?

Cheers,

Nalin Pithwa

### Two cute problems in HP : IITJEE Foundations\Mains, pre RMO

Problem 1:

If $a^{2}, b^{2}, c^{2}$ are in AP, show that $b+c, c+a, a+b$ are in HP.

Proof 1:

Note that a straight forward proof is not so easy.

Below is a nice clever solution:

By adding $ab+bc+ca$ to each term, we see that:

$a^{2}+ab+ac+bc, b^{2}+ab+ac+bc, c^{2}+ab+ac+bc$ are in AP.

that is, $(a+b)(a+c), (b+c)(b+a), (c+a)(c+b)$ are in AP.

Dividing each term by $(a+b)(b+c)(c+a)$.

$\frac{1}{b+c}, \frac{1}{c+a}, \frac{1}{a+b}$ are in AP.

that is, $b+c, c+a, a+b$ are in HP.

QED.

Problem 2:

If the $p^{th}, q^{th}, r^{th}, s^{th}$ terms of an AP are in GP, show that $p-q, q-r, r-s$ are in GP.

Proof 2:

Once again a straight forward proof is not at all easy.

Below is a “bingo” sort of proof 🙂

With the usual notation, we have

$\frac{a+(p-1)d}{a+(q-1)d} = \frac{a+(q-1)d}{a+(r-1)d} = \frac{a+(r-1)d}{a+(s-1)d}$

Hence, each of the ratios is equal to

$\frac{(a+(p-1)d)-(a+(q-1)d)}{(a+(q-1)d)-(a+(r-1)d)} = \frac{(a+(q-1)d)-(a+(r-1)d)}{(a+(r-1)d)-(a+(s-1)d)}$

which in turn is equal  to $\frac{p-q}{q-r} = \frac{q-r}{r-s}$

Hence, $p-q, q-r, r-s$ are in GP.

Cheers,

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, 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