Math topic中文

Conditions for existence

Conditions for the existence of inverse functions and composite functions.

Practice questions

  1. 1The functions ff and gg are defined by f:x↦16−4x,  x∈R,  x≤4,f : x \mapsto \sqrt{16 - 4x},\; x \in \mathbb{R},\; x \le 4, g:x↦x2,  x∈R.g : x \mapsto x^{2},\; x \in \mathbb{R}.…
  2. 2The function g\text{g} is defined by g:x→sin⁡x+x\text{g} : x \to \sin x + x, for x∈Rx \in \mathbb{R}, −π≤x≤π-\pi \le x \le \pi. Show…
  3. 3The function gg is defined by g:x↦ln⁡(x−3)g : x \mapsto \ln(x - \sqrt{3}), for x∈Rx \in \mathbb{R}, x>3x > \sqrt{3}. Explain how yo…
  4. 4The function ff is defined by f:x↦x2−4x, x∈R, ∣x∣<1.f : x \mapsto x^2 - 4x,\ x \in \mathbb{R},\ |x|<1. The function gg is defined by g:x↦1+1x+3, x∈R, x≠−3.g : x \mapsto 1 + \frac{1}{x+3},\ x \in \mathbb{R},\ x \ne -3.…
  5. 5The function f, with domain the set of all real values, is given by f(x)={4−x2for 0<x≤2,2x−4for 2<x≤4,\text{f}(x) = \begin{cases} 4 - x^2 & \text{for } 0 < x \le 2, \\ 2x - 4 & \text{for } 2 < x \le 4, \end{cases}…
  6. 6The functions f, g and h are defined as follows: f : x→sin⁡(πx),x∈R,0≤x≤2x \rightarrow \sin(\pi x), x \in \mathbb{R}, 0 \le x \le 2, g : x→ax,x∈R,x≥−1x \rightarrow a^{x}, x \in \mathbb{R}, x \ge -1…
  7. 7The functions f and g are defined by f:x↦∣(x+1)(x−3)∣,x∈R,f: x \mapsto |(x+1)(x-3)|,\quad x \in \mathbb{R}, g:x↦x−2,x∈R and x≥−1.g: x \mapsto x-2,\quad x \in \mathbb{R}\text{ and }x \ge -1.…
  8. 8Functions f and g are defined by f:x↦[ln⁡(x−1)]2+2,x≥a,f : x \mapsto [\ln(x-1)]^{2} + 2,\quad x \ge a, g:x↦4+3x−x2,x≤32.g : x \mapsto 4 + 3x - x^{2},\quad x \le \frac{3}{2}.…
  9. 9The function f is defined by f:x↦11−x2f: x \mapsto \dfrac{1}{1 - x^2}, −1<x≤0-1 < x \le 0. (i) Define f−1f^{-1}. (ii) Sketch the gra…
  10. 10A function g is defined by g(x)={21+x2if x≤0,0if x>0.g(x)=\begin{cases} \displaystyle\frac{2}{1+x^2} &\text{if } x\le 0,\\[4pt] 0 &\text{if } x>0. \end{cases}…
  11. 11The function f\mathrm{f} is defined by f:x↦5+2x\mathrm{f} : x \mapsto \sqrt{5+2x}, for x∈Rx \in \mathbb{R}, x>−52x > -\frac{5}{2}…
  12. 12The functions f and g are defined by f:x↦x2+3x+1, x∈R, x≤−32,f : x \mapsto x^2 + 3x + 1,\ x \in \mathbb{R},\ x \le -\frac{3}{2}, g:x↦3+e−x, x∈R.g : x \mapsto 3 + e^{-x},\ x \in \mathbb{R}.…
  13. 13It is given that f(x)={12x2,for 1≤x<2,14(−3x+14),for 2≤x<4,f(x) = \begin{cases} \tfrac{1}{2}x^2, & \text{for } 1 \le x < 2, \\ \tfrac{1}{4}(-3x+14), & \text{for } 2 \le x < 4, \end{cases}…
  14. 14The function f is defined by f:x↦ex+12x+2for x∈R, x≠−1.f : x \mapsto e^x + \dfrac{1}{2x+2}\quad\text{for } x \in \mathbb{R},\ x \ne -1. A func…
  15. 15The function f\mathrm{f} is defined by f(x)={4−xfor 1≤x<3,(x−4)2for 3≤x<4,\mathrm{f}(x) = \begin{cases} 4 - x & \text{for } 1 \le x < 3, \\ (x - 4)^2 & \text{for } 3 \le x < 4, \end{cases}…
  16. 16The function f\text{f} is defined by f:x↦1x2−x−6+2,x∈R, x≠−2, x≠3.\text{f} : x \mapsto \frac{1}{x^2 - x - 6} + 2, \quad x \in \mathbb{R}, \ x \ne -2, \ x \ne 3.…
  17. 17The function f\text{f} is defined by f:x→2x−12x,0<x<2.\text{f} : x \to 2x - \frac{1}{2x}, \quad 0 < x < 2. It is given that f−1\text{f}^{-1}…
  18. 18Functions f\mathrm{f} and g\mathrm{g} are defined by f:x→4(x−4)2,x∈R,x≠4,\mathrm{f} : x \to \frac{4}{(x-4)^2}, \quad x \in \mathbb{R}, x \ne 4,…
  19. 19The function f is defined by f:x↦4+1(x−1)2+1,x≥0f : x \mapsto 4 + \dfrac{1}{(x-1)^2 + 1},\quad x \ge 0. The function g is defined by g :…
  20. 20The function g\text{g} is defined by g:x↦ln⁡(x−3)\text{g} : x \mapsto \ln(x - \sqrt{3}), for x∈Rx \in \mathbb{R}, x>3x > \sqrt{3}.…
  21. 21The function f\mathrm{f} is defined by f:x↦axx−a,for x∈R, −a≤x<a,\mathrm{f} : x \mapsto \frac{ax}{x-a}, \quad \text{for } x \in \mathbb{R}, \ -a \le x < a,…
  22. 22The functions f and g are defined by f:x↦1−1x2−1f : x \mapsto 1-\dfrac{1}{x^{2}-1} for x∈R,  x<−1x \in \mathbb{R},\; x<-1, g:x↦x2+4x+5g : x \mapsto x^{2}+4x+5…
  23. 23Complete the square for 2x2−9x+52x^2 - 9x + 5. In the rest of the question, the domain of f\mathrm{f} is x∈Rx \in \mathbb{R},…
  24. 24The function f is defined by f:x↦{a2x−ax2for x≤12a,−12a2x+12a3for x>12a.f : x \mapsto \begin{cases} a^2 x - a x^2 & \text{for } x \le \tfrac{1}{2}a, \\ -\tfrac{1}{2}a^2 x + \tfrac{1}{2}a^3 & \text{for } x > \tfrac{1}{2}a. \end{cases}…
  25. 25The function f is defined as follows: f(x)=x+1x−a,a<x≤bf(x)=x+\frac{1}{x-a},\qquad a<x\le b where a is a positive constant. Given tha…
  26. 26The function f\mathrm{f} is defined by f:x↦2xx2−1,x∈R, x>1\mathrm{f} : x \mapsto \frac{2x}{x^2 - 1}, \quad x \in \mathbb{R}, \ x > 1. Sk…
  27. 27A function h\text{h} is self-inverse if h(x)=h−1(x)\text{h}(x) = \text{h}^{-1}(x) for all xx in the domain of h\text{h}. The f…
  28. 28A function g is defined by g(x)={21+x2if x≤0,0if x>0.g(x)=\begin{cases} \dfrac{2}{1+x^2} &\text{if } x \le 0, \\ 0 &\text{if } x > 0. \end{cases}…
  29. 29The function f\mathrm{f} is defined by f:x↦1−4x2, for x∈R,x<−1.\mathrm{f} : x \mapsto 1 - \frac{4}{x^2}, \text{ for } x \in \mathbb{R}, x < -1.…
  30. 30The function f is defined by f:x↦12−x2,x∈R,  x≤0,  x≠−2 .f:x\mapsto\dfrac{1}{2-x^2},\quad x\in\mathbb{R},\;x\le 0,\;x\ne -\sqrt{2}\,. Define the…
  31. 31The functions f\mathrm{f} and g\mathrm{g} are defined by f:x↦2x+143x−2,x∈R,x>23,g:x↦sin⁡xx,x∈R,x>0.\begin{aligned} \mathrm{f} &: x \mapsto \frac{2x+14}{3x-2}, \quad x \in \mathbb{R}, x > \frac{2}{3}, \\ \mathrm{g} &: x \mapsto \frac{\sin x}{x}, \quad x \in \mathbb{R}, x > 0. \end{aligned}…
  32. 32The function f\text{f} is defined by f:x↦axx−a,for x∈R, −a≤x<a,\text{f} : x \mapsto \frac{ax}{x - a}, \quad \text{for } x \in \mathbb{R}, \ -a \le x < a,…
  33. 33The function f\mathrm{f} is defined by f:x↦cos⁡(πx2)\mathrm{f} : x \mapsto \cos\left(\frac{\pi x}{2}\right), x∈Rx \in \mathbb{R}, −2<x≤0-2 < x \le 0…
  34. 34The function g is defined by g:x↦ln⁡(x−3)g : x \mapsto \ln\bigl(x - \sqrt{3}\bigr), for x∈Rx \in \mathbb{R}, x>3x > \sqrt{3}. Explai…
  35. 35The function f\mathrm{f} is defined by f:x↦ln⁡x2−(ln⁡x)2,x>0.\mathrm{f} : x \mapsto \ln x^2 - (\ln x)^2, \quad x > 0. Sketch the graph of…
  36. 36The function f\text{f} is defined by f:x→13(ex−2−1),x>2\text{f} : x \to \frac{1}{3}(e^{x-2}-1), x > 2. Find f−1\text{f}^{-1} and write d…
  37. 37The function f is defined as follows. f:x↦x2−3xf : x \mapsto x^2 - 3x, for ∣x∣<a|x| < a. (i) State the largest value of a for whic…
  38. 38The function f\mathrm{f} is given by f(x)={52−x,x∈R, x<21x,x∈R, x≥2\mathrm{f}(x) = \begin{cases} \frac{5}{2} - x, & x \in \mathbb{R},\ x < 2 \\ \frac{1}{x}, & x \in \mathbb{R},\ x \ge 2 \end{cases}…
  39. 39The functions f and g are defined below. f(x)={22−xfor 0≤x<2,ln⁡(x−2)for 2<x≤4.f(x)=\begin{cases} \dfrac{2}{2-x} & \text{for } 0\le x<2, \\ \ln(x-2) & \text{for } 2<x\le 4. \end{cases}…
  40. 40The department of statistics of a country has developed two mathematical functions to analyse the foreign worker policy.…
  41. 41The function f is defined as follows. f : x−>−1x2+1x -> \displaystyle -\frac{1}{x^{2}+1} for x∈Rx \in \mathbb{R}. Sketch the gra…
  42. 42The functions f and g are defined by f:x↦∣(x+1)(x−3)∣,x∈R,f:x\mapsto |(x+1)(x-3)|,\quad x\in\mathbb{R}, g:x↦x−2,x∈R and x≥−1.g:x\mapsto x-2,\quad x\in\mathbb{R}\ \text{and } x\ge -1.…
  43. 43The function f\mathrm{f} is defined by f:x↦x−1x\mathrm{f} : x \mapsto \frac{x-1}{x}, x∈Rx \in \mathbb{R}, x≠0x \ne 0 and x≠1x \ne 1…
  44. 44The function ff is defined by f:x↦x2−4xf : x \mapsto x^2 - 4x, x∈Rx \in \mathbb{R}, ∣x∣<1|x| < 1. Show that f−1f^{-1} exists and fi…
  45. 45The function f is defined by f:x↦e(x+c)2f : x \mapsto e^{(x+c)^2}, x∈Rx \in \mathbb{R}, x≤kx \le k, c>0c>0. Find the largest value o…
  46. 46Functions f and g are defined by f:x↦4(x−4)2,x∈R, x≠4,f : x \mapsto \dfrac{4}{(x-4)^2},\quad x \in \mathbb{R},\ x \ne 4, g:x↦ln⁡ ⁣(1+1x),x∈R, x>0.g : x \mapsto \ln\!\left(1 + \dfrac{1}{x}\right),\quad x \in \mathbb{R},\ x > 0.…
  47. 47The functions f and g are defined by f:x↦x2+2x+4,  x≤−1f : x \mapsto x^{2} + 2x + 4, \; x \le -1 and g:x↦ln⁡(2x−1),  x>12g : x \mapsto \ln(2x-1), \; x > \tfrac{1}{2}…
  48. 48The functions f\mathrm{f} and g\mathrm{g} are defined by f:x↦∣4+2x−x2∣, x∈R, x≥3.5,\mathrm{f} : x \mapsto \left| 4 + 2x - x^2 \right|, \ x \in \mathbb{R}, \ x \ge 3.5,…
  49. 49The function f is defined by f : x↦2x2−8x+11x \mapsto 2x^2 - 8x + 11 for x∈Rx \in \mathbb{R}. (i) Express f(x) in the form a(x+b)2+ca(x+b)^2 + c…
  50. 50The functions f and g are defined by f:xightsquigarrow⌊2x+2λx−2⌋,x∈R, −λ<x<2f : x ightsquigarrow \left\lfloor\dfrac{2x+2\lambda}{x-2}\right\rfloor, \quad x \in \mathbb{R}, \ -\lambda < x < 2…
  51. 51The functions f and g are defined as follows: f:x↦∣ex−3−1∣,x≤3f : x \mapsto |e^{x-3}-1|,\quad x \le 3 g:x↦a2−x2,x∈R,  x<0g : x \mapsto a^2 - x^2,\quad x \in \mathbb{R},\; x < 0…
  52. 52The function f\mathrm{f} is defined by f:x↦x2+4x−5,for x≤k,k∈R.\mathrm{f} : x \mapsto x^2 + 4x - 5, \quad \text{for } x \le k, k \in \mathbb{R}.…
  53. 53Section A: Pure Mathematics [40 marks] The function f\text{f} is defined by f:x↦2xx−2\text{f} : x \mapsto \frac{2x}{x-2},…
  54. 54The functions f\text{f} and g\text{g} are defined as follows f:x↦ln⁡(3x−2)x,x>23\text{f} : x \mapsto \frac{\ln(3x - 2)}{x}, x > \frac{2}{3}…
  55. 55The functions f\text{f} and g\text{g} are defined by f:x↦12e1−x2, x∈R, x≤1 and\text{f} : x \mapsto \frac{1}{2}\text{e}^{1-x^2}, \ x \in \mathbb{R}, \ x \le 1 \text{ and}…
  56. 56The function ff is defined by f:x↦2x−12x,0<x<2f: x \mapsto 2x - \dfrac{1}{2x},\quad 0 < x < 2. It is given that f−1f^{-1} exists. Defi…
  57. 57The functions f\mathrm{f} and g\mathrm{g} are defined by f:x↦x2+2x+4,x≤−1andg:x↦ln⁡(2x−1),x>12.\mathrm{f} : x \mapsto x^2 + 2x + 4, \quad x \le -1 \quad \text{and} \quad \mathrm{g} : x \mapsto \ln(2x - 1), \quad x > \frac{1}{2}.…
  58. 58The function f\text{f} is defined by f:x↦2x2+5x−3,x∈R,x≤a.\text{f} : x \mapsto 2x^2 + 5x - 3, \quad x \in \mathbb{R}, \quad x \le a. Sta…
  59. 59The function f\text{f} is defined by f:x↦x2+4x\text{f} : x \mapsto \frac{x^2 + 4}{x}, x∈Rx \in \mathbb{R}. Give a reason why f\text{f}…
  60. 60The functions f and g are defined by f:x↦∣4+2x−x2∣,  x∈R,  x≥3.5,f : x \mapsto |4+2x - x^{2}|, \; x \in \mathbb{R}, \; x \ge 3.5, g:x↦4+eax,  x∈R,  x≥−1,g : x \mapsto 4+e^{ax}, \; x \in \mathbb{R}, \; x \ge -1,…