{"id":21077,"date":"2020-12-04T12:23:40","date_gmt":"2020-12-04T12:23:40","guid":{"rendered":"https:\/\/samacheer-kalvi.com\/?p=21077"},"modified":"2021-12-06T15:58:41","modified_gmt":"2021-12-06T10:28:41","slug":"samacheer-kalvi-11th-maths-guide-chapter-1-ex-1-3","status":"publish","type":"post","link":"https:\/\/samacheer-kalvi.com\/samacheer-kalvi-11th-maths-guide-chapter-1-ex-1-3\/","title":{"rendered":"Samacheer Kalvi 11th Maths Guide Chapter 1 Sets, Relations and Functions Ex 1.3"},"content":{"rendered":"

Tamilnadu State Board New Syllabus\u00a0Samacheer Kalvi 11th Maths Guide<\/a> Pdf Chapter 1 Sets, Relations and Functions Ex 1.3 Text Book Back Questions and Answers, Notes.<\/p>\n

Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 1 Sets, Relations and Functions Ex 1.3<\/h2>\n

Question 1.
\nSuppose that 120 students are studying in 4 sections of eleventh standard in a school. Let A denote the set of students and B denote the set of the sections. Define a relation from A to B as \u201cx related to y if the student x belongs to the section y\u201d. Is this relation a function? What can you say about the inverse relation? Explain your answer.
\nAnswer:
\nGiven A denotes the set of students and B denotes the set of sections. Aslo given there 120 students and 4 sections.<\/p>\n

Let f be a relation from A to B as \u201cx related to y if the student x belongs to the section y\u201d
\n\"Samacheer
\nTwo are more students in A may belong to same section in B. But one student in A cannot belong to two or more sections in B. Every student in A can belong to any one of the section in B. Therefore \/ is a function.<\/p>\n

In B we can have sections without students. Every element in B need not have preimage in A.
\n\u2234 f need not be onto.
\nThus, f is a function and inverse relation for f need not exist.<\/p>\n

\"Samacheer<\/p>\n

Question 2.
\nWrite the values of f at – 4, 1, -2, 7, 0 if
\n\"Samacheer
\nAnswer:
\n\"Samacheer
\nWhen x = -4
\nf(x) = – x + 4
\nf(-4) = – (-4) + 4
\n= 4 + 4 = 8<\/p>\n

When x = 1
\nf(x) = x – x2<\/sup>
\nf(1) = 1 – 12<\/sup>
\n= 1 – 1 = 0<\/p>\n

\"Samacheer<\/p>\n

When x = -2
\nf(x) = x2<\/sup> – x
\nf(-2) = (-2)2<\/sup> – (-2)
\n= 4 + 2 = 6<\/p>\n

When x – 7
\nf(x) = 0
\n\u21d2 f(7) = 0<\/p>\n

When x = 0
\nf(x) = x2<\/sup> – x
\n\u21d2 f(0) = 02<\/sup> – 0 = 0<\/p>\n

Question 3.
\nWrite the values of f at – 3, 5, 2, – 1, 0 if
\n\"Samacheer
\nAnswer:
\n\"Samacheer<\/p>\n

When x = – 3
\nf(x) = x2<\/sup> + x – 5
\nf(-3) = (-3)2<\/sup> + (-3) – 5
\n= 9 – 3 – 5
\n= 9 – 8 = 1<\/p>\n

When x = 5
\nf(x) = x2<\/sup> + 3x – 2
\nf(5) = 52<\/sup> + 3(5) – 2
\n= 25 + 15 – 2
\n= 40 – 2 = 38<\/p>\n

\"Samacheer<\/p>\n

When x = 2
\nf(x) = x2<\/sup> – 3
\nf(2) = 22<\/sup> – 3 = 4 – 3 = 1<\/p>\n

When x = – 1
\nf(x) = x2<\/sup> + x – 5
\nf(-1) = (-1)2<\/sup> – 1 – 5 = 1 – 1 – 5 = – 5<\/p>\n

When x = 0
\nf(x) = x2<\/sup> – 3
\nf(0) = 02<\/sup> – 3<\/p>\n

Question 4.
\nState whether the following relations are functions or not. If it is a function, check for one – to – oneness and ontoness. If it is not a function, state why?
\n(i) If A = { a, b, c } and f = { (a, c), (b, c), (c, b) }; (f : A \u2192 A)
\nAnswer:
\nA = { a, b, c }
\nf = {(a, c), (b, c), (c, b)}; f : A \u2192 A
\n\"Samacheer
\nf is a function since every element in the domain has a unique image in the codomain.
\nf is not one-one.
\na, b belonging to the domain A has the same image in the codomain A. f is not onto since belonging to the codomain A does not have preimage in the domain A Thus the relation \/ is a function from A to A and it is neither one-one nor onto.<\/p>\n

(ii) If X = { x, y, z } and f = { (x, y), (x, z), (z, x) }; (f: X \u2192 X)
\nAnswer:
\nX = { x, y, z }
\nf = {(x, y), (x, z), (z , x) } f : X \u2192 X
\n\"Samacheer
\nThe relation f: X \u2192 X is not a function since the element x in the domain has two images in the co-domain.<\/p>\n

\"Samacheer<\/p>\n

Question 5.
\nLet A = {1, 2, 3, 4} and B = { a, b , c, d } Give a function from A \u2192 B for each of the following.
\n(i) neither one-to-one nor onto.
\nAnswer:
\n\"Samacheer
\nf = { (1, b) , (2, c) , (3, d) , (4, d)
\nf is a function, it not one to one and not onto.<\/p>\n

(ii) not one – to – one but onto
\nAnswer:
\nDoes not exists<\/p>\n

(iii) one – to – one but not onto
\nAnswer:
\nDoes not exist<\/p>\n

(iv) one – to – one and onto
\nAnswer:
\n\"Samacheer
\nf = { (1, a) , (2, b) , (3, c) , (4, d) }
\nf is a function which is one – to – one and onto.<\/p>\n

\"Samacheer<\/p>\n

Question 6.
\nFind the domain of \\(\\frac{1}{1-2 \\sin x}\\)
\nAnswer:
\nLet f(x) = \\(\\frac{1}{1-2 \\sin x}\\)
\nWhen 1 – 2 sin x = 0
\n\u21d2 1 = 2 sin x
\nsin x = \\(\\frac{1}{2}\\)
\n\u21d2 sin x = sin \\(\\left(\\frac{\\pi}{6}\\right)\\)
\nx = n\u03c0 + (- 1)n<\/sup>\\(\\frac{\\pi}{6}\\), n \u2208 Z
\nsin x = sin \u03b1 \u21d2 x = n\u03c0 + (-1)n<\/sup>d, n \u2208 Z
\n\u2234 Domain of f(x) is
\n\"Samacheer<\/p>\n

Question 7.
\nFind the largest possible domain of the real valued function f(x) = \\(\\frac{\\sqrt{4-x^{2}}}{\\sqrt{x^{2}-9}}\\)
\nAnswer:
\n\"Samacheer
\n\u2234 For no real values of x, f (x) is defined.
\n\u2234 Domain of f(x) = { }<\/p>\n

\"Samacheer<\/p>\n

Question 8.
\nFind the range of the function \\(\\frac{1}{2 \\cos x-1}\\)
\nAnswer:
\nLet f(x) = \\(\\frac{1}{2 \\cos x-1}\\)
\nRange of cosine function is
\n– 1 \u2264 cos x \u2264 1
\n– 2 \u2264 2 cos x \u2264 2
\n– 1 \u2264 2 cos x – 1 \u2264 2 – 1
\n– 3 \u2264 2 cos x – 1 \u2264 1
\n\\(-\\frac{1}{3}\\) \u2265 \\(\\frac{1}{2 \\cos x-1}\\) \u2265 1
\n\"Samacheer<\/p>\n

Question 9.
\nShow that the relation xy = – 2 is a function for a suitable domain. Find the domain and the range of the function.
\nAnswer:
\nxy = – 2 \u21d2 y = -2\/x
\nwhich is a function
\nThe domain is (-\u221e, 0) \u222a (0, \u221e) and range is R – {0}<\/p>\n

\"Samacheer<\/p>\n

Question 10.
\nIf f, g : R \u2192 R are defined by f(x) = |x| + x and g(x) = |x| – x find gof and fog.
\nAnswer:
\nGiven
\n\"Samacheer<\/p>\n

\"Samacheer<\/p>\n

Question 11.
\nIf f, g, h are real-valued functions defined on R, then prove that (f + g)oh = foh + goh what can you say about fo(g + h )? Justify your answer.
\nAnswer:
\nGiven f : R \u2192 R , g : R \u2192 R and h : R \u2192 R (f + g) oh: R \u2192 R and (f o h + g o h) : R \u2192 R for any x \u2208 R.
\n[(f + g)oh] (x) = (f + g) h(x)
\n= f(h(x)) + g(h(x))
\n= foh(x) + goh(x)
\n\u2234 (f + g)oh = foh + goh
\nAlso fo(g + h)(x) = f((g + h)(x)) for any x \u2208 R
\n= f(g(x) + h(x))
\n= f(g(x)) + f(h(x))
\n= fog (x) + foh(x)
\n\u2234 fo(g + h) = fog +foh<\/p>\n

Question 12.
\nIf f : R \u2192 R is defined by f( x ) = 3x – 5, Prove that f is a bijection and find its inverse.
\nAnswer:
\nGiven f(x) = 3x – 5
\nLet y = 3x – 5
\ny + 5 = 3x
\n\u21d2 \\(\\frac{y+5}{3}\\) = x
\nLet g(y) = \\(\\frac{y+5}{3}\\)
\ngof (x) = g(f(x))
\n= g(3x – 5)
\n= \\(\\frac{3 x-5+5}{3}\\) = \\(\\frac{3 x}{3}\\) = x
\ngof (x) = x
\nfog (y) = f(g(y))
\n= f\\(\\left(\\frac{y+5}{3}\\right)\\)
\n= 3\\(\\left(\\frac{y+5}{3}\\right)\\) – 5
\n= y + 5 – 5
\nfog(y) = y
\n\u2234 gof = Ix<\/sub> and fog = IY<\/sub>
\nHence f and g are bijections and inverses to each ot1er.
\nHence f is a bijection and f-1<\/sup>(y) = \\(\\frac{y+5}{3}\\)
\nReplacing y by x we get f-1<\/sup>(x) = \\(\\frac{x+5}{3}\\)<\/p>\n

\"Samacheer<\/p>\n

Question 13.
\nThe weight of the muscles of a man is a function of his bodyweight x and can be expressed as W ( x ) = 0.35x. Determine the domain of this function.
\nAnswer:
\nW(x) = 0.35x
\nSince bodyweight x is positive and if it increases then W(x) also increases.
\nDomain is (0, \u221e) i.e.,x > 0<\/p>\n

Question 14.
\nThe distance of an object falling is a function of time t and can be expressed as s ( t) = – 16t2<\/sup>. Graph the function and determine if it is one – to – one.
\nAnswer:
\nGiven s (t) = – 16t2<\/sup>
\ns (t1<\/sub>) = s (t2<\/sub>) \u21d2 – 16t1<\/sub>2<\/sup> = – 16t2<\/sub>2<\/sup>
\n\u21d2 t1<\/sub>2<\/sup> = t2<\/sub>2<\/sup>
\n\u21d2 \u00b1 t1<\/sub> = \u00b1 t2<\/sub>
\nSince s (t1<\/sub>) = s (t1<\/sub>) \"Samacheer 14 t1<\/sub> = t2<\/sub>
\n\u2234 The function s(t) is not one-one
\nGraph of s(t) = – 16t2<\/sup>
\nTake the time along x – axis and distance along y – axis.
\n\"Samacheer<\/p>\n

\"Samacheer<\/p>\n

Question 15.
\nThe total cost of airfare on a given route is comprised of the base cost C and the fuel Surcharge S in rupee. Both C and S are functions of the mileage m; C ( m ) = 0.4 m + 50 and S ( m ) = 0.03m. Determine a function for the total cost of a ticket in terms of the mileage and find the airfare for flying 1600 miles.
\nAnswer:
\nC – base cost,
\nS = fuel surcharge,
\nm = mileage
\nC(m) = 0.4 m + 50
\nS(m) = 0.03 m
\nTotal cost = C(m) + S(m)
\n= 0.4 m + 50 + 0.03 m
\n= 0.43 m + 50
\nfor 1600 miles
\nT(c) = 0.43 (1600) + 50 = 688 + 50 = \u20b9 738<\/p>\n

Question 16.
\nA salesperson whose annual earnings can be represented by the function A (x) = 30,000 + 0.04 x, where x is the rupee value of the merchandise, he sells. His son is also in sales and his earnings are represented by the function S(x) = 25,000 + 0.05 x. Find (A + S)(x) and determine the total family income if they each sell Rs. 1,50,00,000 worth of merchandise.
\nAnswer:
\nGiven A (x) = 30,000 + 0.04 x
\nS (x) = 25,000 + 0.05x
\nA(x) + S(x) = 30,000 + 0.04 x + 25,000 + 0.05x
\n(A + S)(x) = 55,000 + 0.09 x
\nGiven x = 1,50,00,000
\nThen (A + S)(x) = 55000 + 0.09 \u00d7 1,50,00,000
\n= 55000 + 1,35000000
\nTotal family income = Rs. 14,05,000<\/p>\n

\"Samacheer<\/p>\n

Question 17.
\nThe function for exchanging American dollars for Singapore Dollar on a given day is f (x) = 1.23x, where x represents the number of American dollars. On the same day, the function for exchanging Singapore Dollar to Indian Rupee is g(y) = 50.50y, where y represents the number of Singapore dollars. Write a function which will give the exchange rate of American dollars in terms of Indian rupee.
\nAnswer:
\nGiven f(x) = 1.23x
\nwhere x represents the number of American dollars
\ng(y) = 50.50y
\nwhere y represents the number of Singapore dollars.
\n\"Samacheer
\nTo convert American dollars to Indian rupees, we must find
\ngof (x) = g(f(x))
\n= g (1.23x)
\n= 50.50 (1.23x)
\n= 62.115x
\n\u2234 The function for the exchange rate of American can dollars in terms of Indian rupees is
\ngof (x) = 62.1 15x<\/p>\n

Question 18.
\nThe owner of a small restaurant can prepare a particular meal at a cost of Rs. 100. He estimates that if the menu price of the meal is x rupees, then the number of customers who will order that meal at that price in an evening is given by the function D (x) = 200 – x. Express his day revenue total cost and profit on this meal as functions of x.
\nAnswer:
\ncost of one meal = \u20b9 100
\nTotal cost = \u20b9 100 (200 – x)
\nNumber of customers = 200 – x
\nDay revenue = \u20b9 (200 – x) x
\nTotal profit = day revenue – total cost
\n= (200 – x) x – (100) (200 – x)<\/p>\n

\"Samacheer<\/p>\n

Question 19.
\nThe formula for converting from Fahrenheit to Celsius temperature is y = \\(\\). Find the inverse of this function and determine whether the inverse is also a function?
\nAnswer:
\n\"Samacheer
\n\"Samacheer<\/p>\n

\"Samacheer<\/p>\n

Question 20.
\nA simple cipher takes a number and codes it, using the function f( x) = 3x – 4. Find the inverse of this function, determine whether the inverse is also a function and verify the symmetrical property about the line y = x (by drawing the lines)
\nAnswer:
\nGiven f(x) = 3x – 4
\nLet y = 3x – 4
\n\u21d2 y + 4 = 3x
\n\u21d2 x = \\(\\frac{y+4}{3}\\)
\nLet g(y) = \\(\\frac{y+4}{3}\\)
\ngof (x) = g (f(x) )
\n= g(3x – 4)
\n= \\(\\frac{3 x-4+4}{3}=\\frac{3 x}{3}\\)
\ngof(x) = x
\nand fog(y) = f(g(y))
\n= f\\(\\left(\\frac{y+4}{3}\\right)\\)
\n= 3\\(\\left(\\frac{y+4}{3}\\right)\\)
\n= y + 4 – 4 = y
\nfog (y) = y
\nHence g of = Ix<\/sub> and fog = Iy<\/sub>
\nThis shows that f and g are bijections and inverses of each other.
\nHence f is bijection and f-1<\/sup>(y) = \\(\\frac{y+4}{3}\\)
\nReplacing y by x we get f-1<\/sup>(x) = \\(\\frac{x+4}{3}\\)
\nThe line y = x
\n\"Samacheer
\nf(x) =
\nThe line y =3x-4
\nWhen x = 0 \u21d2 y = 3 \u00d7 0 – 4 = -4
\nWhen x = 1 \u21d2 y = 3 \u00d7 1 – 4 = -1
\nWhen x = -1 \u21d2 y = 3 \u00d7 -1 – 4 = -7
\nWhen x = 2 \u21d2 y = 3 \u00d7 2 – 4 = 2
\nWhen x = -2 \u21d2 y = 3 \u00d7 -2 – 4 = -10
\nWhen x = 3 \u21d2 y = 3 \u00d7 3 – 4 = 5
\nWhen x = -3 \u21d2 y = 3 \u00d7 -3 – 4 = -13
\n\"Samacheer<\/p>\n

\"Samacheer<\/p>\n

The line y = \\(\\frac{x+4}{3}\\)
\n\"Samacheer
\n\"Samacheer<\/p>\n","protected":false},"excerpt":{"rendered":"

Tamilnadu State Board New Syllabus\u00a0Samacheer Kalvi 11th Maths Guide Pdf Chapter 1 Sets, Relations and Functions Ex 1.3 Text Book Back Questions and Answers, Notes. Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 1 Sets, Relations and Functions Ex 1.3 Question 1. Suppose that 120 students are studying in 4 sections of eleventh standard in a …<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[6],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/samacheer-kalvi.com\/wp-json\/wp\/v2\/posts\/21077"}],"collection":[{"href":"https:\/\/samacheer-kalvi.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/samacheer-kalvi.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/samacheer-kalvi.com\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/samacheer-kalvi.com\/wp-json\/wp\/v2\/comments?post=21077"}],"version-history":[{"count":1,"href":"https:\/\/samacheer-kalvi.com\/wp-json\/wp\/v2\/posts\/21077\/revisions"}],"predecessor-version":[{"id":48160,"href":"https:\/\/samacheer-kalvi.com\/wp-json\/wp\/v2\/posts\/21077\/revisions\/48160"}],"wp:attachment":[{"href":"https:\/\/samacheer-kalvi.com\/wp-json\/wp\/v2\/media?parent=21077"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/samacheer-kalvi.com\/wp-json\/wp\/v2\/categories?post=21077"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/samacheer-kalvi.com\/wp-json\/wp\/v2\/tags?post=21077"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}