Mathematics · Textbook solutions

Functions

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 252 questions

6.1 Function

19 q

Worked Examples

Worked · 19
  1. 6.1 SolvedEx.1
    Evaluate f(x)=2x2−3x+4f(x) = 2x^2 - 3x + 4 at x=7x = 7 and x=−2tx = -2t.
  2. 6.1 SolvedEx.2
    Using the graph of y=g(x)y = g(x) given below, find g(−4)g(-4) and g(3)g(3). (In the figure each grid square is 2 units along both axes. The curve rises from the lower left through the marked points (−6, −4)(-6,\,-4) and (−4, 0)(-4,\,0) to a local maximum at the marked point (−1, 4)(-1,\,4), then falls to a local minimum near x=3x = 3, and rises again, crossing the X-axis near x=9x = 9.)
  3. 6.1 SolvedEx.3
    If t(m)=3m2−mt(m) = 3m^2 - m and t(m)=4t(m) = 4, then find mm.
  4. 6.1 SolvedEx.4
    From the graph below find xx for which f(x)=4f(x) = 4. (Fig. 6.15 shows y=f(x)y = f(x), an upward-opening parabola whose vertex touches the X-axis at (1, 0)(1,\,0), drawn on a unit grid with the X-axis marked from −2-2 to 55 and the Y-axis marked 11 to 55, together with the horizontal line y=4y = 4.)
  5. From the equation 4x+7y=14x + 7y = 1 express
    6.1 SolvedEx.5(i)
    yy as a function of xx.
  6. 6.1 SolvedEx.5(ii)
    xx as a function of yy.
  7. 6.1 SolvedEx.6
    Find the domain and range of f(x)=9−x2f(x) = \sqrt{9 - x^2}.
  8. 6.1 SolvedEx.7
    Find the domain of f(x)=x3−8f(x) = \sqrt{x^3 - 8}.
  9. 6.1 SolvedEx.8
    Find domain and range of the function f(x)=6−4x24x+5f(x) = \frac{6 - 4x^2}{4x + 5}.
  10. 6.1 SolvedEx.9
    Solve 52x+7=1255^{2x+7} = 125.
  11. 6.1 SolvedEx.10
    Find the domain of f(x)=6−2x−23−xf(x) = \sqrt{6 - 2^{x} - 2^{3-x}}.
  12. 6.1 SolvedEx.11
    Write log⁡72\log 72 in terms of log⁡2\log 2 and log⁡3\log 3.
  13. 6.1 SolvedEx.12
    Evaluate ln⁡e9−ln⁡e4\ln e^{9} - \ln e^{4}.
  14. 6.1 SolvedEx.13
    Expand log⁡[x3(x+3)2(x−4)2]\log\left[\frac{x^{3}(x+3)}{2(x-4)^{2}}\right].
  15. 6.1 SolvedEx.14
    Combine 3ln⁡(p+1)−12ln⁡r+5ln⁡(2q+3)3\ln (p + 1) - \frac{1}{2}\ln r + 5\ln(2q + 3) into single logarithm.
  16. 6.1 SolvedEx.15
    Find the domain of ln⁡(x−5)\ln(x - 5).
  17. 6.1 SolvedEx.16
    Evaluate log⁡481log⁡49\frac{\log_{4} 81}{\log_{4} 9}.
  18. 6.1 SolvedEx.17
    Prove that, 2log⁡ba4⋅log⁡cb3⋅log⁡ac5=1202\log_{b} a^{4} \cdot \log_{c} b^{3} \cdot \log_{a} c^{5} = 120.
  19. 6.1 SolvedEx.18
    Find the domain of f(x)=log⁡x+5(x2−4)f(x) = \log_{x+5}(x^{2} - 4).

Exercise 6.1

84 q
  1. Check if the following relations are functions.
    Ex 6.1 Q1(a)
    The relation shown in Fig. 6.32, from A={2,1,0,−1,−2}A = \{2, 1, 0, -1, -2\} to B={−3,2,1,5,6,−1}B = \{-3, 2, 1, 5, 6, -1\}, in which 22 is joined to −3-3, 11 is joined to 11, 00 is joined to 22, −1-1 is joined to 11 and −2-2 is joined to 55.
  2. Ex 6.1 Q1(b)
    The relation shown in Fig. 6.33, from A={p,q,r,s}A = \{p, q, r, s\} to B={a,b,c,d,e}B = \{a, b, c, d, e\}, in which p→ap \to a, q→cq \to c, r→br \to b, r→dr \to d and s→es \to e.
  3. Ex 6.1 Q1(c)
    The relation shown in Fig. 6.34, from A={3,−2,1,0,2,4}A = \{3, -2, 1, 0, 2, 4\} to B={9,7,−6,3,2}B = \{9, 7, -6, 3, 2\}, in which 3→73 \to 7, 1→91 \to 9, 0→−60 \to -6, 2→32 \to 3 and 4→24 \to 2; the element −2-2 of AA is joined to nothing.
  4. Which sets of ordered pairs represent functions from A={1,2,3,4}A = \{1, 2, 3, 4\} to B={−1,0,1,2,3}B = \{-1, 0, 1, 2, 3\}? Justify.
    Ex 6.1 Q2(a)
    {(1,0),(3,3),(2,−1),(4,1),(2,2)}\{(1,0), (3,3), (2,-1), (4,1), (2,2)\}
  5. Ex 6.1 Q2(b)
    {(1,2),(2,−1),(3,1),(4,3)}\{(1,2), (2,-1), (3,1), (4,3)\}
  6. Ex 6.1 Q2(c)
    {(1,3),(4,1),(2,2)}\{(1,3), (4,1), (2,2)\}
  7. Ex 6.1 Q2(d)
    {(1,1),(2,1),(3,1),(4,1)}\{(1,1), (2,1), (3,1), (4,1)\}
  8. Check if the relation given by the equation represents yy as function of xx.
    Ex 6.1 Q3(a)
    2x+3y=122x + 3y = 12
  9. Ex 6.1 Q3(b)
    x+y2=9x + y^2 = 9
  10. Ex 6.1 Q3(c)
    x2−y=25x^2 - y = 25
  11. Ex 6.1 Q3(d)
    2y+10=02y + 10 = 0
  12. Ex 6.1 Q3(e)
    3x−6=213x - 6 = 21
  13. If f(m)=m2−3m+1f(m) = m^2 - 3m + 1, find
    Ex 6.1 Q4(a)
    f(0)f(0)
  14. Ex 6.1 Q4(b)
    f(−3)f(-3)
  15. Ex 6.1 Q4(c)
    f(12)f\left(\frac{1}{2}\right)
  16. Ex 6.1 Q4(d)
    f(x+1)f(x + 1)
  17. Ex 6.1 Q4(e)
    f(−x)f(-x)
  18. Ex 6.1 Q4(f)
    (f(2+h)−f(2)h)\left(\frac{f(2+h) - f(2)}{h}\right), h≠0h \ne 0.
  19. Find xx, if g(x)=0g(x) = 0 where
    Ex 6.1 Q5(a)
    g(x)=5x−67g(x) = \frac{5x - 6}{7}
  20. Ex 6.1 Q5(b)
    g(x)=18−2x27g(x) = \frac{18 - 2x^2}{7}
  21. Ex 6.1 Q5(c)
    g(x)=6x2+x−2g(x) = 6x^2 + x - 2
  22. Ex 6.1 Q5(d)
    g(x)=x3−2x2−5x+6g(x) = x^3 - 2x^2 - 5x + 6
  23. Find xx, if f(x)=g(x)f(x) = g(x) where
    Ex 6.1 Q6(a)
    f(x)=x4+2x2f(x) = x^4 + 2x^2, g(x)=11x2g(x) = 11x^2
  24. Ex 6.1 Q6(b)
    f(x)=x−3f(x) = \sqrt{x} - 3, g(x)=5−xg(x) = 5 - x
  25. Ex 6.1 Q7
    If f(x)=a−xb−xf(x) = \frac{a - x}{b - x}, f(2)f(2) is undefined, and f(3)=5f(3) = 5, find aa and bb.
  26. Find the domain and range of the following functions.
    Ex 6.1 Q8(a)
    f(x)=7x2+4x−1f(x) = 7x^2 + 4x - 1
  27. Ex 6.1 Q8(b)
    g(x)=x+4x−2g(x) = \frac{x + 4}{x - 2}
  28. Ex 6.1 Q8(c)
    h(x)=x+55+xh(x) = \frac{\sqrt{x + 5}}{5 + x}
  29. Ex 6.1 Q8(d)
    f(x)=x+13f(x) = \sqrt[3]{x + 1}
  30. Ex 6.1 Q8(e)
    f(x)=(x−2)(5−x)f(x) = \sqrt{(x - 2)(5 - x)}
  31. Ex 6.1 Q8(f)
    f(x)=x−37−xf(x) = \sqrt{\frac{x - 3}{7 - x}}
  32. Ex 6.1 Q8(g)
    f(x)=16−x2f(x) = \sqrt{16 - x^2}
  33. Express the area AA of a square as a function of its
    Ex 6.1 Q9(a)
    side ss
  34. Ex 6.1 Q9(b)
    perimeter PP
  35. Express the area AA of circle as a function of its
    Ex 6.1 Q10(a)
    radius rr
  36. Ex 6.1 Q10(b)
    diameter dd
  37. Ex 6.1 Q10(c)
    circumference CC
  38. Ex 6.1 Q11
    An open box is made from a square of cardboard of 30 cms side, by cutting squares of length xx centimeters from each corner and folding the sides up. Express the volume of the box as a function of xx. Also find its domain.
  39. Ex 6.1 Q12
    Let ff be a subset of Z×ZZ \times Z defined by f={(ab,a+b):a,b∈Z}f = \{(ab, a + b) : a, b \in Z\}. Is ff a function from ZZ to ZZ? Justify.
  40. Check the injectivity and surjectivity of the following functions.
    Ex 6.1 Q13(a)
    f:N→Nf : N \to N given by f(x)=x2f(x) = x^2
  41. Ex 6.1 Q13(b)
    f:Z→Zf : Z \to Z given by f(x)=x2f(x) = x^2
  42. Ex 6.1 Q13(c)
    f:R→Rf : R \to R given by f(x)=x2f(x) = x^2
  43. Ex 6.1 Q13(d)
    f:N→Nf : N \to N given by f(x)=x3f(x) = x^3
  44. Ex 6.1 Q13(e)
    f:R→Rf : R \to R given by f(x)=x3f(x) = x^3
  45. Ex 6.1 Q14
    Show that if f:A→Bf : A \to B and g:B→Cg : B \to C are one-one, then g∘fg \circ f is also one-one.
  46. Ex 6.1 Q15
    Show that if f:A→Bf : A \to B and g:B→Cg : B \to C are onto, then g∘fg \circ f is also onto.
  47. Ex 6.1 Q16
    If f(x)=3(4x+1)f(x) = 3(4^{x+1}) find f(−3)f(-3).
  48. Express the following exponential equations in logarithmic form
    Ex 6.1 Q17(a)
    25=322^5 = 32
  49. Ex 6.1 Q17(b)
    540=154^0 = 1
  50. Ex 6.1 Q17(c)
    231=2323^1 = 23
  51. Ex 6.1 Q17(d)
    93/2=279^{3/2} = 27
  52. Ex 6.1 Q17(e)
    3−4=1813^{-4} = \frac{1}{81}
  53. Ex 6.1 Q17(f)
    10−2=0.0110^{-2} = 0.01
  54. Ex 6.1 Q17(g)
    e2=7.3890e^2 = 7.3890
  55. Ex 6.1 Q17(h)
    e1/2=1.6487e^{1/2} = 1.6487
  56. Ex 6.1 Q17(i)
    e−x=6e^{-x} = 6
  57. Express the following logarithmic equations in exponential form
    Ex 6.1 Q18(a)
    log⁡264=6\log_2 64 = 6
  58. Ex 6.1 Q18(b)
    log⁡5125=−2\log_5 \frac{1}{25} = -2
  59. Ex 6.1 Q18(c)
    log⁡100.001=−3\log_{10} 0.001 = -3
  60. Ex 6.1 Q18(d)
    log⁡1/2(−8)=3\log_{1/2} (-8) = 3
  61. Ex 6.1 Q18(e)
    ln⁡1=0\ln 1 = 0
  62. Ex 6.1 Q18(f)
    ln⁡e=1\ln e = 1
  63. Ex 6.1 Q18(g)
    ln⁡12=−0.693\ln \frac{1}{2} = -0.693
  64. Find the domain of
    Ex 6.1 Q19(a)
    f(x)=ln⁡(x−5)f(x) = \ln(x - 5)
  65. Ex 6.1 Q19(b)
    f(x)=log⁡10(x2−5x+6)f(x) = \log_{10}(x^2 - 5x + 6)
  66. Write the following expressions as sum or difference of logarithms
    Ex 6.1 Q20(a)
    log⁡(pqrs)\log\left(\frac{pq}{rs}\right)
  67. Ex 6.1 Q20(b)
    log⁡(x y3)\log\left(\sqrt{x}\,\sqrt[3]{y}\right)
  68. Ex 6.1 Q20(c)
    ln⁡(a3(a−2)2b2+5)\ln\left(\frac{a^3 (a - 2)^2}{\sqrt{b^2 + 5}}\right)
  69. Ex 6.1 Q20(d)
    ln⁡[x−23 (2x+1)4(x+4)2x+4]2\ln\left[\frac{\sqrt[3]{x - 2}\,(2x + 1)^4}{(x + 4)\sqrt{2x + 4}}\right]^2
  70. Write the following expressions as a single logarithm.
    Ex 6.1 Q21(a)
    5log⁡x+7log⁡y−log⁡z5\log x + 7\log y - \log z
  71. Ex 6.1 Q21(b)
    13log⁡(x−1)+12log⁡(x)\frac{1}{3}\log(x - 1) + \frac{1}{2}\log(x)
  72. Ex 6.1 Q21(c)
    ln⁡(x+2)+ln⁡(x−2)−3ln⁡(x+5)\ln(x + 2) + \ln(x - 2) - 3\ln(x + 5)
  73. Ex 6.1 Q22
    Given that log⁡2=a\log 2 = a and log⁡3=b\log 3 = b, write log⁡96\log \sqrt{96} in terms of aa and bb.
  74. Prove that
    Ex 6.1 Q23(a)
    blog⁡ba=ab^{\log_b a} = a
  75. Ex 6.1 Q23(b)
    log⁡bma=1mlog⁡ba\log_{b^m} a = \frac{1}{m}\log_b a
  76. Ex 6.1 Q23(c)
    alog⁡cb=blog⁡caa^{\log_c b} = b^{\log_c a}
  77. Ex 6.1 Q24
    If f(x)=ax2−bx+6f(x) = ax^2 - bx + 6 and f(2)=3f(2) = 3 and f(4)=30f(4) = 30, find aa and bb.
  78. Solve for xx.
    Ex 6.1 Q25(a)
    log⁡2+log⁡(x+3)−log⁡(3x−5)=log⁡3\log 2 + \log(x + 3) - \log(3x - 5) = \log 3
  79. Ex 6.1 Q25(b)
    2log⁡10x=1+log⁡10(x+1110)2\log_{10} x = 1 + \log_{10}\left(x + \frac{11}{10}\right)
  80. Ex 6.1 Q25(c)
    log⁡2x+log⁡4x+log⁡16x=214\log_2 x + \log_4 x + \log_{16} x = \frac{21}{4}
  81. Ex 6.1 Q25(d)
    x+log⁡10(1+2x)=xlog⁡105+log⁡106x + \log_{10}(1 + 2^x) = x\log_{10} 5 + \log_{10} 6
  82. Ex 6.1 Q26
    If log⁡(x+y3)=12log⁡x+12log⁡y\log\left(\frac{x + y}{3}\right) = \frac{1}{2}\log x + \frac{1}{2}\log y, show that xy+yx=7\frac{x}{y} + \frac{y}{x} = 7.
  83. Ex 6.1 Q27
    If log⁡(x−y4)=log⁡x+log⁡y\log\left(\frac{x - y}{4}\right) = \log \sqrt{x} + \log \sqrt{y}, show that (x+y)2=20xy(x + y)^2 = 20xy.
  84. Ex 6.1 Q28
    If x=log⁡abcx = \log_a bc, y=log⁡bcay = \log_b ca, z=log⁡cabz = \log_c ab then prove that 11+x+11+y+11+z=1\frac{1}{1 + x} + \frac{1}{1 + y} + \frac{1}{1 + z} = 1.

6.2 Algebra of Functions

22 q

Worked Examples

Worked · 22
  1. If f(x)=x2+2f(x) = x^2 + 2 and g(x)=5x−8g(x) = 5x - 8, then find
    6.2 SolvedEx.1(i)
    (f+g)(1)(f + g)(1)
  2. 6.2 SolvedEx.1(ii)
    (f−g)(−2)(f - g)(-2)
  3. 6.2 SolvedEx.1(iii)
    (f∘g)(3m)(f \circ g)(3m)
  4. 6.2 SolvedEx.1(iv)
    fg(0)\dfrac{f}{g}(0)
  5. Given the function f(x)=5x2f(x) = 5x^2 and g(x)=4−xg(x) = \sqrt{4 - x} find the domain of
    6.2 SolvedEx.2(i)
    (f+g)(x)(f + g)(x)
  6. 6.2 SolvedEx.2(ii)
    (f∘g)(x)(f \circ g)(x)
  7. 6.2 SolvedEx.2(iii)
    fg(x)\dfrac{f}{g}(x)
  8. If f(x)=2x+5f(x) = \dfrac{2}{x + 5} and g(x)=x2−1g(x) = x^2 - 1, then find
    6.2 SolvedEx.3(i)
    (f∘g)(x)(f \circ g)(x)
  9. 6.2 SolvedEx.3(ii)
    (g∘f)(3)(g \circ f)(3)
  10. 6.2 SolvedEx.4
    If f(x)=x2f(x) = x^2, g(x)=x+5g(x) = x + 5, and h(x)=1xh(x) = \dfrac{1}{x}, x≠0x \ne 0, find (g∘f∘h)(x)(g \circ f \circ h)(x)
  11. 6.2 SolvedEx.5
    If h(x)=(x−5)2h(x) = (x - 5)^2, find the functions ff and gg, such that h=f∘gh = f \circ g.
  12. 6.2 SolvedEx.6
    Express m(x)=1x3+7m(x) = \dfrac{1}{x^3 + 7} in the form of f∘g∘hf \circ g \circ h
  13. 6.2 SolvedEx.7
    If ff is one-one onto function with f(x)=9−5xf(x) = 9 - 5x, find f−1(−1)f^{-1}(-1).
  14. 6.2 SolvedEx.8
    Verify that f(x)=x−58f(x) = \dfrac{x - 5}{8} and g(x)=8x+5g(x) = 8x + 5 are inverse functions of each other.
  15. 6.2 SolvedEx.9
    Determine whether the function f(x)=2x+1x−3f(x) = \dfrac{2x + 1}{x - 3} has inverse, if it exists find it.
  16. 6.2 SolvedEx.10
    Solve ∣4x−5∣≤3|4x-5|\le 3.
  17. 6.2 SolvedEx.11
    Find the domain of 1 ∣∣x∣−1∣−3 \dfrac{1}{\sqrt{\,\big||x|-1\big|-3\,}}.
  18. 6.2 SolvedEx.12
    Solve ∣x−1∣+∣x+2∣=8|x-1|+|x+2|=8.
  19. 6.2 SolvedEx.13
    Evaluate {5.2}+{−5.2}\{5.2\}+\{-5.2\} and {7}+{−7}\{7\}+\{-7\}, where {x}\{x\} is the fractional part function.
  20. 6.2 SolvedEx.14
    Evaluate {2.8+5}\{2.8+5\} and {2.8−5}\{2.8-5\}, and compare each with {2.8}\{2.8\}, where {x}\{x\} is the fractional part function.
  21. 6.2 SolvedEx.15
    If {x}\{x\} and [x][x] are the fractional part function and greatest integer function of xx respectively, solve for xx, if {x+1}+2x=4[x+1]−6\{x+1\}+2x=4[x+1]-6.
  22. 6.2 SolvedEx.16
    Given that log⁡102=0.3010\log_{10}2=0.3010, find the number of digits in the number 201020^{10}.

Exercise 6.2

45 q
  1. If f(x)=3x+5f(x)=3x+5, g(x)=6x−1g(x)=6x-1, then find
    Ex 6.2 Q1(a)
    (f+g)(x)(f+g)(x)
  2. Ex 6.2 Q1(b)
    (f−g)(2)(f-g)(2)
  3. Ex 6.2 Q1(c)
    (fg)(3)(fg)(3)
  4. Ex 6.2 Q1(d)
    (fg)(x)\left(\dfrac{f}{g}\right)(x) and its domain.
  5. Ex 6.2 Q2
    Let f:{2,4,5}→{2,3,6}f:\{2,4,5\}\to\{2,3,6\} and g:{2,3,6}→{2,4}g:\{2,3,6\}\to\{2,4\} be given by f={(2,3),(4,6),(5,2)}f=\{(2,3),(4,6),(5,2)\} and g={(2,4),(3,4),(6,2)}g=\{(2,4),(3,4),(6,2)\}. Write down g∘fg\circ f.
  6. If f(x)=2x2+3f(x)=2x^{2}+3, g(x)=5x−2g(x)=5x-2, then find
    Ex 6.2 Q3(a)
    f∘gf\circ g
  7. Ex 6.2 Q3(b)
    g∘fg\circ f
  8. Ex 6.2 Q3(c)
    f∘ff\circ f
  9. Ex 6.2 Q3(d)
    g∘gg\circ g
  10. Verify that ff and gg are inverse functions of each other, where
    Ex 6.2 Q4(a)
    f(x)=x−74f(x)=\dfrac{x-7}{4}, g(x)=4x+7g(x)=4x+7
  11. Ex 6.2 Q4(b)
    f(x)=x3+4f(x)=x^{3}+4, g(x)=x−43g(x)=\sqrt[3]{x-4}
  12. Ex 6.2 Q4(c)
    f(x)=x+3x−2f(x)=\dfrac{x+3}{x-2}, g(x)=2x+3x−1g(x)=\dfrac{2x+3}{x-1}
  13. Check if the following functions have an inverse function. If yes, find the inverse function.
    Ex 6.2 Q5(a)
    f(x)=5x2f(x)=5x^{2}
  14. Ex 6.2 Q5(b)
    f(x)=8f(x)=8
  15. Ex 6.2 Q5(c)
    f(x)=6x−73f(x)=\dfrac{6x-7}{3}
  16. Ex 6.2 Q5(d)
    f(x)=4x+5f(x)=\sqrt{4x+5}
  17. Ex 6.2 Q5(e)
    f(x)=9x3+8f(x)=9x^{3}+8
  18. Ex 6.2 Q5(f)
    f(x)={x+7x<08−xx≥0f(x)=\begin{cases} x+7 & x<0 \\ 8-x & x\ge 0 \end{cases}
  19. If f(x)={x2+3,x≤25x+7,x>2f(x)=\begin{cases} x^{2}+3, & x\le 2 \\ 5x+7, & x>2 \end{cases}, then find
    Ex 6.2 Q6(a)
    f(3)f(3)
  20. Ex 6.2 Q6(b)
    f(2)f(2)
  21. Ex 6.2 Q6(c)
    f(0)f(0)
  22. If f(x)={4x−2,x≤−35,−3<x<3x2,x≥3f(x)=\begin{cases} 4x-2, & x\le -3 \\ 5, & -3<x<3 \\ x^{2}, & x\ge 3 \end{cases}, then find
    Ex 6.2 Q7(a)
    f(−4)f(-4)
  23. Ex 6.2 Q7(b)
    f(−3)f(-3)
  24. Ex 6.2 Q7(c)
    f(1)f(1)
  25. Ex 6.2 Q7(d)
    f(5)f(5)
  26. If f(x)=2∣x∣+3xf(x)=2|x|+3x, then find
    Ex 6.2 Q8(a)
    f(2)f(2)
  27. Ex 6.2 Q8(b)
    f(−5)f(-5)
  28. If f(x)=4[x]−3f(x)=4[x]-3, where [x][x] is greatest integer function of xx, then find
    Ex 6.2 Q9(a)
    f(7.2)f(7.2)
  29. Ex 6.2 Q9(b)
    f(0.5)f(0.5)
  30. Ex 6.2 Q9(c)
    f(−52)f\left(-\dfrac{5}{2}\right)
  31. Ex 6.2 Q9(d)
    f(2π)f(2\pi), where π=3.14\pi=3.14
  32. If f(x)=2{x}+5xf(x)=2\{x\}+5x, where {x}\{x\} is fractional part function of xx, then find
    Ex 6.2 Q10(a)
    f(−1)f(-1)
  33. Ex 6.2 Q10(b)
    f(14)f\left(\dfrac{1}{4}\right)
  34. Ex 6.2 Q10(c)
    f(−1.2)f(-1.2)
  35. Ex 6.2 Q10(d)
    f(−6)f(-6)
  36. Solve the following for xx, where ∣x∣|x| is modulus function, [x][x] is greatest integer function, [x][x] is a fractional part function.
    Ex 6.2 Q11(a)
    ∣x+4∣≥5|x+4|\ge 5
  37. Ex 6.2 Q11(b)
    ∣x−4∣+∣x−2∣=3|x-4|+|x-2|=3
  38. Ex 6.2 Q11(c)
    x2+7∣x∣+12=0x^{2}+7|x|+12=0
  39. Ex 6.2 Q11(d)
    ∣x∣≤3|x|\le 3
  40. Ex 6.2 Q11(e)
    2∣x∣=52|x|=5
  41. Ex 6.2 Q11(f)
    [x+[x+[x]]]=9\big[x+[x+[x]]\big]=9
  42. Ex 6.2 Q11(g)
    {x}>4\{x\}>4
  43. Ex 6.2 Q11(h)
    {x}=0\{x\}=0
  44. Ex 6.2 Q11(i)
    {x}=0.5\{x\}=0.5
  45. Ex 6.2 Q11(j)
    2{x}=x+[x]2\{x\}=x+[x]

Miscellaneous Exercise 6

82 q

(I) Select the correct answer

Practice · 10
  1. Misc I Q1
    If log⁡(5x−9)−log⁡(x+3)=log⁡2\log(5x-9)-\log(x+3)=\log 2 then x=x= ...............
    1. A.
      3
    2. B.
      5
    3. C.
      2
    4. D.
      7
  2. Misc I Q2
    If log⁡10(log⁡10(log⁡10x))=0\log_{10}\left(\log_{10}\left(\log_{10}x\right)\right)=0 then x=x=
    1. A.
      1000
    2. B.
      101010^{10}
    3. C.
      10
    4. D.
      0
  3. Misc I Q3
    Find xx, if 2log⁡2x=42\log_{2}x=4
    1. A.
      4, −44,\ -4
    2. B.
      4
    3. C.
      −4-4
    4. D.
      not defined
  4. Misc I Q4
    The equation log⁡x216+log⁡2x64=3\log_{x^{2}}16+\log_{2x}64=3 has,
    1. A.
      one irrational solution
    2. B.
      no prime solution
    3. C.
      two real solutions
    4. D.
      one integral solution
  5. Misc I Q5
    If f(x)=11−xf(x)=\frac{1}{1-x}, then f(f{f(x)}]f(f\{f(x)\}] is
    1. A.
      x−1x-1
    2. B.
      1−x1-x
    3. C.
      xx
    4. D.
      −x-x
  6. Misc I Q6
    If f:R→Rf:\mathbb{R}\to\mathbb{R} is defined by f(x)=x3f(x)=x^{3} then f−1(8)f^{-1}(8) is equal to :
    1. A.
      {2}\{2\}
    2. B.
      {−2. 2}\{-2.\ 2\}
    3. C.
      {−2}\{-2\}
    4. D.
      (−2. 2)(-2.\ 2)
  7. Misc I Q7
    Let the function ff be defined by f(x)=2x+11−3xf(x)=\frac{2x+1}{1-3x} then f−1(x)f^{-1}(x) is:
    1. A.
      x−13x+2\frac{x-1}{3x+2}
    2. B.
      x+13x−2\frac{x+1}{3x-2}
    3. C.
      2x+11−3x\frac{2x+1}{1-3x}
    4. D.
      3x+2x−1\frac{3x+2}{x-1}
  8. Misc I Q8
    If f(x)=2x2+bx+cf(x)=2x^{2}+bx+c and f(0)=3f(0)=3 and f(2)=1f(2)=1, then f(1)f(1) is equal to
    1. A.
      −2-2
    2. B.
      0
    3. C.
      1
    4. D.
      2
  9. Misc I Q9
    The domain of 1[x]−x\frac{1}{[x]-x} where [x][x] is greatest integer function is
    1. A.
      R\mathbb{R}
    2. B.
      Z\mathbb{Z}
    3. C.
      R−Z\mathbb{R}-\mathbb{Z}
    4. D.
      Q−{0}\mathbb{Q}-\{0\}
  10. Misc I Q10
    The domain and range of f(x)=2−∣x−5∣f(x)=2-|x-5| is
    1. A.
      R+, (−∞,1]\mathbb{R}^{+},\ (-\infty,1]
    2. B.
      R, (−∞,2]\mathbb{R},\ (-\infty,2]
    3. C.
      R, (−∞,2)\mathbb{R},\ (-\infty,2)
    4. D.
      R+, (−∞,2]\mathbb{R}^{+},\ (-\infty,2]

(II) Answer the following

Practice · 72
  1. Which of the following relations are functions? If it is a function determine its domain and range.
    Misc II Q1(i)
    {(2,1),(4,2),(6,3),(8,4),(10,5),(12,6),(14,7)}\{(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)\}
  2. Misc II Q1(ii)
    {(0,0),(1,1),(1,−1),(4,2),(4,−2),(9,3),(9,−3),(16,4),(16,−4)}\{(0, 0), (1, 1), (1, -1), (4, 2), (4, -2), (9, 3), (9, -3), (16, 4), (16, -4)\}
  3. Misc II Q1(iii)
    {2,1),(3,1),(5,2)}\{2, 1), (3, 1), (5, 2)\}
  4. Find whether following functions are one-one.
    Misc II Q2(i)
    f:R→Rf : \mathrm{R} \to \mathrm{R} defined by f(x)=x2+5f(x) = x^2 + 5
  5. Misc II Q2(ii)
    f:R−{3}→Rf : \mathrm{R} - \{3\} \to \mathrm{R} defined by f(x)=5x+7x−3f(x) = \frac{5x + 7}{x - 3} for x∈R−{3}x \in \mathrm{R} - \{3\}
  6. Find whether following functions are onto or not.
    Misc II Q3(i)
    f:Z→Zf : \mathrm{Z} \to \mathrm{Z} defined by f(x)=6x−7f(x) = 6x - 7 for all x∈Zx \in \mathrm{Z}
  7. Misc II Q3(ii)
    f:R→Rf : \mathrm{R} \to \mathrm{R} defined by f(x)=x2+3f(x) = x^2 + 3 for all x∈Rx \in \mathrm{R}
  8. Misc II Q4
    Let f:R→Rf : \mathrm{R} \to \mathrm{R} be a function defined by f(x)=5x3−8f(x) = 5x^3 - 8 for all x∈Rx \in \mathrm{R}, show that ff is one-one and onto. Hence find f−1f^{-1}.
  9. Misc II Q5
    A function f:R→Rf : \mathrm{R} \to \mathrm{R} defined by f(x)=3x5+2f(x) = \frac{3x}{5} + 2, x∈Rx \in \mathrm{R}. Show that ff is one-one and onto. Hence find f−1f^{-1}.
  10. Misc II Q6
    A function ff is defined as f(x)=4x+5f(x) = 4x + 5, for −4≤x<0-4 \le x < 0. Find the values of f(−1)f(-1), f(−2)f(-2), f(0)f(0), if they exist.
  11. A function ff is defined as : f(x)=5−xf(x) = 5 - x for 0≤x≤40 \le x \le 4. Find the value of xx such that
    Misc II Q7(i)
    f(x)=3f(x) = 3
  12. Misc II Q7(ii)
    f(x)=5f(x) = 5
  13. Misc II Q8
    If f(x)=3x4−5x2+7f(x) = 3x^4 - 5x^2 + 7 find f(x−1)f(x - 1).
  14. Misc II Q9
    If f(x)=3x+af(x) = 3x + a and f(1)=7f(1) = 7 find aa and f(4)f(4).
  15. Misc II Q10
    If f(x)=ax2+bx+2f(x) = ax^2 + bx + 2 and f(1)=3f(1) = 3, f(4)=42f(4) = 42, find aa and bb.
  16. Find composite of ff and gg
    Misc II Q11(i)
    f={(1,3),(2,4),(3,5),(4,6)}f = \{(1, 3), (2, 4), (3, 5), (4, 6)\}, g={(3,6),(4,8),(5,10),(6,12)}g = \{(3, 6), (4, 8), (5, 10), (6, 12)\}
  17. Misc II Q11(ii)
    f={(1,1),(2,4),(3,4),(4,3)}f = \{(1, 1), (2, 4), (3, 4), (4, 3)\}, g={(1,1),(3,27),(4,64)}g = \{(1, 1), (3, 27), (4, 64)\}
  18. Find fogfog and gofgof
    Misc II Q12(i)
    f(x)=x2+5f(x) = x^2 + 5, g(x)=x−8g(x) = x - 8
  19. Misc II Q12(ii)
    f(x)=3x−2f(x) = 3x - 2, g(x)=x2g(x) = x^2
  20. Misc II Q12(iii)
    f(x)=256x4f(x) = 256x^4, g(x)=xg(x) = \sqrt{x}
  21. Misc II Q13
    If f(x)=2x−15x−2f(x) = \frac{2x - 1}{5x - 2}, x≠52x \ne \frac{5}{2} Show that (fof)(x)=x(fof)(x) = x.
  22. Misc II Q14
    If f(x)=x+34x−5f(x) = \frac{x + 3}{4x - 5}, g(x)=3+5x4x−1g(x) = \frac{3 + 5x}{4x - 1} then show that (fog)(x)=x(fog)(x) = x.
  23. Misc II Q15
    Let f:R−{2}→Rf : \mathrm{R} - \{2\} \to \mathrm{R} be defined by f(x)=x2−4x−2f(x) = \frac{x^2 - 4}{x - 2} and g:R→Rg : \mathrm{R} \to \mathrm{R} be defined by g(x)=x+2g(x) = x + 2. Ex whether f=gf = g or not.
  24. Misc II Q16
    Let f:R→Rf : \mathrm{R} \to \mathrm{R} be given by f(x)=x+5f(x) = x + 5 for all x∈Rx \in \mathrm{R}. Draw its graph.
  25. Misc II Q17
    Let f:R→Rf : \mathrm{R} \to \mathrm{R} be given by f(x)=x3+1f(x) = x^3 + 1 for all x∈Rx \in \mathrm{R}. Draw its graph.
  26. Misc II Q18
    For any base show that log⁡(1+2+3)=log⁡1+log⁡2+log⁡3\log(1 + 2 + 3) = \log 1 + \log 2 + \log 3.
  27. Misc II Q19
    Find xx, if x=33log⁡32x = 3^{3\log_3 2}
  28. Misc II Q20
    Show that, log⁡∣x2+1+x∣+log⁡∣x2+1−x∣=0\log\left|\sqrt{x^2 + 1} + x\right| + \log\left|\sqrt{x^2 + 1} - x\right| = 0
  29. Misc II Q21
    Show that, log⁡a2bc+log⁡b2ca+log⁡c2ab=0\log\frac{a^2}{bc} + \log\frac{b^2}{ca} + \log\frac{c^2}{ab} = 0
  30. Misc II Q22
    Simplify, log⁡(log⁡x4)−log⁡(log⁡x)\log(\log x^4) - \log(\log x).
  31. Misc II Q23
    Simplify log⁡102845−log⁡1035324+log⁡10325432−log⁡101315\log_{10}\frac{28}{45} - \log_{10}\frac{35}{324} + \log_{10}\frac{325}{432} - \log_{10}\frac{13}{15}
  32. Misc II Q24
    If log⁡(a+b2)=12(log⁡a+log⁡b)\log\left(\frac{a + b}{2}\right) = \frac{1}{2}(\log a + \log b), then show that a=ba = b
  33. Misc II Q25
    If b2=acb^2 = ac. prove that, log⁡a+log⁡c=2log⁡b\log a + \log c = 2\log b
  34. Misc II Q26
    Solve for xx, log⁡x(8x−3)−log⁡x4=2\log_x(8x - 3) - \log_x 4 = 2
  35. Misc II Q27
    If a2+b2=7aba^2 + b^2 = 7ab, show that, log⁡(a+b3)=12log⁡a+12log⁡b\log\left(\frac{a + b}{3}\right) = \frac{1}{2}\log a + \frac{1}{2}\log b
  36. Misc II Q28
    If log⁡(x−y5)=12log⁡x+12log⁡y\log\left(\frac{x - y}{5}\right) = \frac{1}{2}\log x + \frac{1}{2}\log y, show that x2+y2=27xyx^2 + y^2 = 27xy.
  37. Misc II Q29
    If log⁡3[log⁡2(log⁡3x)]=1\log_3[\log_2(\log_3 x)] = 1, show that x=6561x = 6561.
  38. Misc II Q30
    If f(x)=log⁡(1−x)f(x) = \log(1 - x), 0≤x<10 \le x < 1 show that f(11+x)=f(1−x)−f(−x)f\left(\frac{1}{1 + x}\right) = f(1 - x) - f(-x)
  39. Misc II Q31
    Without using log tables, prove that 25<log⁡103<12\frac{2}{5} < \log_{10} 3 < \frac{1}{2}
  40. Misc II Q32
    Show that 7log⁡(1516)+6log⁡(83)+5log⁡(25)+log⁡(3225)=log⁡37\log\left(\frac{15}{16}\right) + 6\log\left(\frac{8}{3}\right) + 5\log\left(\frac{2}{5}\right) + \log\left(\frac{32}{25}\right) = \log 3
  41. Misc II Q33
    Solve : log⁡2x4+4log⁡42x=2\sqrt{\log_2 x^4} + 4\log_4\sqrt{\frac{2}{x}} = 2
  42. Misc II Q34
    Find value of 3+log⁡103432+12log⁡10(494)+12log⁡10(125)\dfrac{3 + \log_{10} 343}{2 + \frac{1}{2}\log_{10}\left(\frac{49}{4}\right) + \frac{1}{2}\log_{10}\left(\frac{1}{25}\right)}
  43. Misc II Q35
    If log⁡ax+y−2z=log⁡by+z−2x=log⁡cz+x−2y\dfrac{\log a}{x + y - 2z} = \dfrac{\log b}{y + z - 2x} = \dfrac{\log c}{z + x - 2y}, show that abc=1abc = 1.
  44. Misc II Q36
    Show that, log⁡yx3⋅log⁡zy4⋅log⁡xz5=60\log_y x^3 \cdot \log_z y^4 \cdot \log_x z^5 = 60
  45. Misc II Q37
    If log⁡2a4=log⁡2b6=log⁡2c3k\dfrac{\log_2 a}{4} = \dfrac{\log_2 b}{6} = \dfrac{\log_2 c}{3k} and a3b2c=1a^3 b^2 c = 1 find the value of kk.
  46. Misc II Q38
    If a2=b3=c4=d5a^2 = b^3 = c^4 = d^5, show that log⁡abcd=4730\log_a bcd = \frac{47}{30}.
  47. Solve the following for xx, where ∣x∣|x| is modulus function, [x][x] is greatest interger function, {x}\{x\} is a fractional part function.
    Misc II Q39(a)
    1<∣x−1∣<41 < |x - 1| < 4
  48. Misc II Q39(b)
    ∣x2−x−6∣=x+2|x^2 - x - 6| = x + 2
  49. Misc II Q39(c)
    ∣x2−9∣+∣x2−4∣=5|x^2 - 9| + |x^2 - 4| = 5
  50. Misc II Q39(d)
    −2<[x]≤7-2 < [x] \le 7
  51. Misc II Q39(e)
    2[2x−5]−1=72[2x - 5] - 1 = 7
  52. Misc II Q39(f)
    [x2]−5[x]+6=0[x^2] - 5[x] + 6 = 0
  53. Misc II Q39(g)
    [x−2]+[x+2]+{x}=0[x - 2] + [x + 2] + \{x\} = 0
  54. Misc II Q39(h)
    [x2]+[x3]=5x6\left[\frac{x}{2}\right] + \left[\frac{x}{3}\right] = \frac{5x}{6}
  55. Find the domain of the following functions.
    Misc II Q40(a)
    f(x)=x2+4x+4x2+x−6f(x) = \frac{x^2 + 4x + 4}{x^2 + x - 6}
  56. Misc II Q40(b)
    f(x)=x−3+1log⁡(5−x)f(x) = \sqrt{x - 3} + \frac{1}{\log(5 - x)}
  57. Misc II Q40(c)
    f(x)=1−1−1−x2f(x) = \sqrt{1 - \sqrt{1 - \sqrt{1 - x^2}}}
  58. Misc II Q40(d)
    f(x)=x!f(x) = x!
  59. Misc II Q40(e)
    f(x)=5−xPx−1f(x) = {}^{5-x}P_{x-1}
  60. Misc II Q40(f)
    f(x)=x−x2+5−xf(x) = \sqrt{x - x^2} + \sqrt{5 - x}
  61. Misc II Q40(g)
    f(x)=log⁡(x2−6x+6)f(x) = \sqrt{\log(x^2 - 6x + 6)}
  62. Find the range of the following functions.
    Misc II Q41(a)
    f(x)=∣x−5∣f(x) = |x - 5|
  63. Misc II Q41(b)
    f(x)=x9+x2f(x) = \frac{x}{9 + x^2}
  64. Misc II Q41(c)
    f(x)=11+xf(x) = \frac{1}{1 + \sqrt{x}}
  65. Misc II Q41(d)
    f(x)=[x]−xf(x) = [x] - x
  66. Misc II Q41(e)
    f(x)=1+2x+4xf(x) = 1 + 2^x + 4^x
  67. Find (f∘g)(x)(f \circ g)(x) and (g∘f)(x)(g \circ f)(x)
    Misc II Q42(a)
    f(x)=exf(x) = e^x, g(x)=log⁡xg(x) = \log x
  68. Misc II Q42(b)
    f(x)=xx+1f(x) = \frac{x}{x + 1}, g(x)=x1−xg(x) = \frac{x}{1 - x}
  69. Find f(x)f(x) if
    Misc II Q43(a)
    g(x)=x2+x−2g(x) = x^2 + x - 2 and (g∘f)(x)=4x2−10x+4(g \circ f)(x) = 4x^2 - 10x + 4
  70. Misc II Q43(b)
    g(x)=1+xg(x) = 1 + \sqrt{x} and f[g(x)]=3+2x+xf[g(x)] = 3 + 2\sqrt{x} + x.
  71. Find (f∘f)(x)(f \circ f)(x) if
    Misc II Q44(a)
    f(x)=x1+x2f(x) = \frac{x}{\sqrt{1 + x^2}}
  72. Misc II Q44(b)
    f(x)=2x+13x−2f(x) = \frac{2x + 1}{3x - 2}