Math topic中文

Inverse and composite

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 curve C has equation x+y=(x−y)2x + y = (x - y)^2. It is given that C has only one turning point. Show that 1−dydx=21+2x−2y1 - \frac{dy}{dx} = \frac{2}{1 + 2x - 2y}…
  3. 3Show that for x>1x>1, ddx(1x2−1)=−x(x2−1)3/2 .\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{1}{\sqrt{x^2-1}}\right)=-\dfrac{x}{(x^2-1)^{3/2}}\,.…
  4. 4The 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…
  5. 5The 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.…
  6. 6The 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}…
  7. 7The 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…
  8. 8下列函数中，满足f(x2)=[f(x)]2f(x^{2})=[f(x)]^{2}的是()() A. f(x)=ln⁡xf(x)=\ln x B. f(x)=∣x+1∣f(x)=|x+1| C. f(x)=x3f(x)=x^{3} D. f(x)=exf(x)=e^{x}
  9. 9A function f is defined as f(x)=ln⁡(5−x),  x<5.f(x)=\ln(5-x),\; x<5. Find the volume of revolution when the region bounded by the curve of…
  10. 10The 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.…
  11. 11Functions 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}.…
  12. 12A 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}…
  13. 13The 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}…
  14. 14The 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}.…
  15. 15It 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}…
  16. 16The 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…
  17. 17已知函数f(x)=ax+b(a>0f(x)=a^{x}+b(a>0且a≠1)a\neq 1)的图像过点(1,7)(1,7)，其反函数y=f−1(x)y=f^{-1}(x)的图像过点(4,0)(4,0)，则aa的值为___.
  18. 18The 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}…
  19. 19The function f is defined by f:x↦1x2+6x+5f : x \mapsto \dfrac{1}{x^2 + 6x + 5}, for x≥−3x \ge -3, x≠−1x \ne -1. The function g is defi…
  20. 20Topic 4: Functions [MI/2014/MYE PU1/13] The function f\mathrm{f} is defined by f:x↦x2−6x+14,x∈R.\mathrm{f} : x \mapsto x^2 - 6x + 14, \quad x \in \mathbb{R}.…
  21. 21The 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.…
  22. 22The 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}…
  23. 23Functions 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,…
  24. 24The function f is defined by f:x↦5(x−4)2f : x \mapsto \dfrac{5}{(x-4)^2} for x∈R, x<4x\in\mathbb{R},\ x<4. Find f−1(x)f^{-1}(x) and state t…
  25. 25The 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 :…
  26. 26The 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}.…
  27. 27The 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,…
  28. 28The 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…
  29. 29Complete 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},…
  30. 30The 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}…
  31. 31The 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…
  32. 32The 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…
  33. 33A 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…
  34. 34A 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}…
  35. 35The graph below shows the graph of y=f(x)y=f(x). Sketch on the same diagram the graph of y=f−1(x)y=f^{-1}(x), clearly stating the…
  36. 36The function f is defined by f:x↦x3−1f : x \mapsto x^{3} - 1, x∈Rx \in \mathbb{R}. (i) Find an expression for f−1(x)f^{-1}(x). (ii)…
  37. 37The 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.…
  38. 38A sequence of real numbers x1,x2,x3,…x_1, x_2, x_3, \dots satisfies the recurrence relation xn+1=5xn+22xn+3 for all n≥1.x_{n+1} = \frac{5x_n + 2}{2x_n + 3} \text{ for all } n \ge 1.…
  39. 39The 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…
  40. 40The 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}…
  41. 41The diagram shows the curve CC with equation y=2sin⁡−1xy = 2\sin^{-1}x and the line LL with equation y=8π3x−πy = \frac{8\pi}{3}x - \pi…
  42. 42The 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,…
  43. 43A function f\mathrm{f} is said to be self-inverse if f(x)=f−1(x)\mathrm{f}(x) = \mathrm{f}^{-1}(x) for all xx in the domain of…
  44. 44The 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…
  45. 45The 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…
  46. 46The 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…
  47. 47The 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…
  48. 48The 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…
  49. 49The diagram shows the function ff defined for 0≤x≤60 \le x \le 6 by f:x↦12x2for 0≤x≤2,f : x \mapsto \tfrac{1}{2}x^{2}\quad\text{for }0 \le x \le 2,…
  50. 50The function h is defined as follows. h:x↦ln⁡(x−k+1), x∈R, x≥k,h : x \mapsto \ln(x - k + 1),\ x \in \mathbb{R},\ x \ge k, where k<0k < 0. Find t…
  51. 51The 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}…
  52. 52The 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}…
  53. 53Let RR be the region bounded by the curves y=4xy = 4\sqrt{x} and y=x2+1y = x^2 + 1. Find the area of region RR. If the line…
  54. 54The department of statistics of a country has developed two mathematical functions to analyse the foreign worker policy.…
  55. 55The function f\text{f} is defined by f:x↦−x−3x+1, for x∈R,x≠k.\text{f} : x \mapsto \frac{-x - 3}{x + 1}, \text{ for } x \in \mathbb{R}, x \ne k.…
  56. 56The 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…
  57. 57The 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.…
  58. 58设 f(x)=2x2−10x+6f(x) = 2x^2 - 10x + 6，x≤kx \leq k。 a 求使 f−1(x)f^{-1}(x) 存在的最大 kk 值。 b 对此 kk 值：i 求 f−1(x)f^{-1}(x)。ii 写出 f−1(x)f^{-1}(x)…
  59. 59The 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…
  60. 60The 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…