Graphical Representation of Functions

Maths, HSSC-Punjab Second Year. This chapter is split into 79 topics and has 1,861 practice questions. The topics covered are listed below, followed by 10 free sample questions from the chapter.

Topics in this chapter

1. Quadratic graph: the perfect-square form 27 questions
2. Logarithmic inequalities: base greater than 1 23 questions
3. Transformations: worked cases from the book re-derived 20 questions
4. Exponential function: positivity, continuity and injectivity arguments 21 questions
5. Growth and decay model: finding the amount after time t 20 questions
6. Growth and decay model: decay constant from data 21 questions
7. Sound intensity: finding the level in decibels 20 questions
8. Sound intensity: ratio of intensities and decibel differences 21 questions
9. Predicting a quadratic from its graph: writing the equation 20 questions
10. Transformations: effect on intercepts, vertex and asymptotes 23 questions
11. Growth and decay model: doubling time and meaning of k 20 questions
12. Inverse hyperbolic functions: derivation steps 21 questions
13. Algebraic and transcendental functions: classification 28 questions
14. Transformations: combined shifts y = f(x - h) + k 28 questions
15. Exponential equations: reduction to a quadratic 23 questions
16. Inverse hyperbolic functions: domain and range 23 questions
17. Laws of logarithms: power law and numerical combinations 22 questions
18. Transformations: vertical scaling y = a f(x) 24 questions
19. Exponential equations: different bases and logarithms 24 questions
20. Logarithmic function: domain of expressions 27 questions
21. Hyperbolic functions: graphs of sinh x and cosh x 21 questions
22. Common and natural logarithms 20 questions
23. Polynomial graphs in factor form: cubics and quartics 22 questions
24. Change of base formula 30 questions
25. Transformations: combinations applied in order 21 questions
26. Exponential function: intercepts, asymptote and range of transformed forms 30 questions
27. Exponential inequalities: different bases and two-term forms 23 questions
28. Hyperbolic functions: definitions 24 questions
29. Logarithms: evaluating by definition 26 questions
30. Logarithmic equations: variable base 22 questions
31. Inverse functions: domain, range and symmetry 22 questions
32. Exponential equations: reducing to a common base 24 questions
33. Exponential function: evaluating powers 24 questions
34. Graph of the inverse function: reflection in the line y = x 24 questions
35. One-to-one functions, horizontal line test and existence of an inverse 21 questions
36. Laws of logarithms: product and quotient 20 questions
37. Growth and decay model: decay time and half-life 20 questions
38. Exponential inequalities: same base 24 questions
39. Exponential function: increasing, decreasing and comparing values 28 questions
40. Logarithmic function: points on the graph and different bases 26 questions
41. Modulus function and its graph 30 questions
42. Inverse hyperbolic functions: evaluating 30 questions
43. Compound interest: continuous compounding time, rate and principal 20 questions
44. Logarithmic equations: equal arguments and domain checks 23 questions
45. Compound interest: finding time, rate and principal (discrete) 21 questions
46. Logarithmic function: definition and exponential form 27 questions
47. Logarithms: basic identities 24 questions
48. Fundamental and non-fundamental transcendental functions 22 questions
49. Logarithmic equations: sum of logarithms 23 questions
50. Quadratic graph from factors: x-intercepts 30 questions
51. Quadratic graph: sketching an expanded quadratic by factorizing 24 questions
52. Logarithmic function: increasing or decreasing by base 20 questions
53. Compound interest: continuous compounding amount 20 questions
54. Transformations: reflections and negative scaling 21 questions
55. Quadratic graph from factors: vertex 26 questions
56. Predicting a quadratic: touching points, vertex and y-intercept data 21 questions
57. Algebraic and transcendental functions: definitions 21 questions
58. Hyperbolic functions: identities with given values 30 questions
59. Hyperbolic functions: domain and range 26 questions
60. Quadratic graph from factors: y-intercept and end behaviour 26 questions
61. Hyperbolic identities: proofs 21 questions
62. Sound intensity and the decibel scale: concepts 20 questions
63. Compound interest: discrete compounding amount 22 questions
64. Transformations: horizontal shift y = f(x - h) 24 questions
65. Predicting a quadratic from its graph: finding the constant a 24 questions
66. Exponential function: key values of e^x and (1/2)^x 25 questions
67. Exponential function: asymptotic behaviour and limits 23 questions
68. Hyperbolic functions: numerical values 24 questions
69. Transformations: vertical shift y = f(x) + k 24 questions
70. Logarithmic inequalities: base between 0 and 1 25 questions
71. Logarithmic function: asymptote, intercepts and continuity 26 questions
72. Sound intensity: finding the intensity from the level 21 questions
73. Logarithmic inequalities: sums of logarithms 20 questions
74. Compound interest: comparing compounding frequencies 20 questions
75. Transformations: horizontal scaling y = f(bx) 26 questions
76. Inverse hyperbolic functions: logarithmic formulas 23 questions
77. Exponential function: definition, base and the number e 22 questions
78. Growth model: finding the time and the constant k 28 questions
79. Finding the inverse of a one-to-one function 30 questions

Sample questions: Graphical Representation of Functions

10 sample questions with the correct answer and an explanation. Try to answer each one before reading the explanation.

Q1. Which one of the following functions is algebraic?

• A. (eˣ − e⁻ˣ)/2
• B. √(x² + 2x)(correct answer)
• C. cot x
• D. tan⁻¹(x²)

Answer: B.√(x² + 2x) is built from x using finitely many algebraic operations (addition, subtraction, multiplication, division, roots), so it is algebraic; the others involve exponential, logarithmic, trigonometric or hyperbolic parts.

Q2. Which of these four functions is an algebraic function?

• A. (x − 1)³(correct answer)
• B. tan x
• C. cos 2x + 1
• D. x ln x

Answer: A.(x − 1)³ is built from x using finitely many algebraic operations (addition, subtraction, multiplication, division, roots), so it is algebraic; the others involve exponential, logarithmic, trigonometric or hyperbolic parts.

Q3. Identify the algebraic function among the following.

• A. cot x
• B. x√x(correct answer)
• C. tan x
• D. x sin x

Answer: B.x√x is built from x using finitely many algebraic operations (addition, subtraction, multiplication, division, roots), so it is algebraic; the others involve exponential, logarithmic, trigonometric or hyperbolic parts.

Q4. Exactly one of these functions is algebraic. Which one?

• A. (eˣ − e⁻ˣ)/2
• B. cot x
• C. (x + 2)/(x − 2)(correct answer)
• D. x ln x

Answer: C.(x + 2)/(x − 2) is built from x using finitely many algebraic operations (addition, subtraction, multiplication, division, roots), so it is algebraic; the others involve exponential, logarithmic, trigonometric or hyperbolic parts.

Q5. Which function below can be built from x by finitely many algebraic operations and roots?

• A. ln 2x
• B. cos 2x + 1
• C. x√x(correct answer)
• D. sec x

Answer: C.x√x is built from x using finitely many algebraic operations (addition, subtraction, multiplication, division, roots), so it is algebraic; the others involve exponential, logarithmic, trigonometric or hyperbolic parts.

Q6. Which of the following functions satisfies a polynomial equation in y with polynomial coefficients in x?

• A. tan⁻¹(x² + 1) + 2ˣ
• B. ln(x + 3) + 2ˣ/(x − 1)
• C. √(2x − 1)/(x + 3)(correct answer)
• D. x sin x + ln(2x − 1)

Answer: C.√(2x − 1)/(x + 3) is built from x using finitely many algebraic operations (addition, subtraction, multiplication, division, roots), so it is algebraic; the others involve exponential, logarithmic, trigonometric or hyperbolic parts.

Q7. Solve log(3x + 2) > log(−2x + 5).

• A. 3/5 ≤ x < 5/2
• B. −2/3 < x < 5/2
• C. 3/5 < x < 5/2(correct answer)
• D. x > 3/5

Answer: C.Domain: 3x + 2 > 0 and −2x + 5 > 0 give −2/3 < x < 5/2. Since log (base 10) is increasing, the inequality becomes 3x + 2 > −2x + 5, i.e. x > 3/5. Combining with the domain gives 3/5 < x < 5/2.

Q8. Solve log(4x − 2) < log(−3x + 8).

• A. 1/2 < x ≤ 10/7
• B. 1/2 < x < 10/7(correct answer)
• C. 1/2 < x < 8/3
• D. x < 10/7

Answer: B.Domain: 4x − 2 > 0 and −3x + 8 > 0 give 1/2 < x < 8/3. Since log (base 10) is increasing, the inequality becomes 4x − 2 < −3x + 8, i.e. x < 10/7. Combining with the domain gives 1/2 < x < 10/7.

Q9. Solve log(2x + 3) < log(−3x + 12).

• A. −3/2 < x < 4
• B. x < 9/5
• C. −3/2 < x ≤ 9/5
• D. −3/2 < x < 9/5(correct answer)

Answer: D.Domain: 2x + 3 > 0 and −3x + 12 > 0 give −3/2 < x < 4. Since log (base 10) is increasing, the inequality becomes 2x + 3 < −3x + 12, i.e. x < 9/5. Combining with the domain gives −3/2 < x < 9/5.

Q10. Solve log(3x − 2) < log(−2x + 10).

• A. x < 12/5
• B. 2/3 < x < 5
• C. 2/3 < x < 12/5(correct answer)
• D. 2/3 < x ≤ 12/5

Answer: C.Domain: 3x − 2 > 0 and −2x + 10 > 0 give 2/3 < x < 5. Since log (base 10) is increasing, the inequality becomes 3x − 2 < −2x + 10, i.e. x < 12/5. Combining with the domain gives 2/3 < x < 12/5.