Math topic中文

Domain restriction for inverse

Domain restriction to obtain an inverse function.

Practice questions

  1. 1The 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}…
  2. 2The 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…
  3. 3The 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.…
  4. 4Functions 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}.…
  5. 5A 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}…
  6. 6The 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}…
  7. 7The 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}.…
  8. 8The 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…
  9. 9A curve CC has equation y=2ax+bxy = 2ax + \frac{b}{x} where aa and bb are non-zero real constants and x≠0x \ne 0. Using diff…
  10. 10The 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…
  11. 11Topic 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}.…
  12. 12Functions 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,…
  13. 13The 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…
  14. 14The 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…
  15. 15The 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}…
  16. 16The 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…
  17. 17A 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}…
  18. 18The 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…
  19. 19The 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…
  20. 20A 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…
  21. 21The 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…
  22. 22The 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…
  23. 23The diagram shows the curve CC with equation y=12(3x+2)2+4y = \frac{12}{(3x+2)^2 + 4} which has a turning point at (−23,3)\left(-\frac{2}{3}, 3\right)…
  24. 24The 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,…
  25. 25The 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…
  26. 26The 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.…
  27. 27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)…
  28. 28The 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…
  29. 29The 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…
  30. 30Functions 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.…
  31. 31The 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,…
  32. 32The 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…
  33. 33The 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}.…
  34. 34The 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}…
  35. 35The 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}…
  36. 36The 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}.…
  37. 37The 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…
  38. 38The 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}…
  39. 39The function ff is defined as f:x↦6x−x2f : x \mapsto 6x - x^{2}. For the rest of this question, the domain of f is assumed to…
  40. 40A curve CC has equation y=ax2+bx+cy = ax^2 + bx + c, where aa, bb and cc are constants. It is given that CC passes through…
  41. 41The 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 fu…
  42. 42The function f\mathrm{f} is given by f:x↦14−(2x+1)2,x∈R, x≤−12, x≠−32.\mathrm{f} : x \mapsto \frac{1}{4 - (2x + 1)^2}, \quad x \in \mathbb{R}, \ x \le -\frac{1}{2}, \ x \ne -\frac{3}{2}.…
  43. 43The function f\text{f} is defined by f:x↦(x−2)2−4,x∈R, x≥0.\text{f} : x \mapsto (x-2)^2 - 4,\quad x \in \mathbb{R},\ x \ge 0. Sketch the…
  44. 44The functions f\mathrm{f} and g\mathrm{g} are defined by f:x↦x−1x,x<0,g:x↦sin⁡x,0≤x≤2π.\begin{aligned} \mathrm{f} &: x \mapsto x - \frac{1}{x}, \quad x < 0, \\ \mathrm{g} &: x \mapsto \sin x, \quad 0 \le x \le 2\pi. \end{aligned}…
  45. 45The functions f\text{f} and g\text{g} are defined by f:x↦x2−2x+5for x∈R,g:x↦1x−1for x∈R, x≠1.\begin{aligned} \text{f} &: x \mapsto x^2 - 2x + 5 \quad \text{for } x \in \mathbb{R}, \\ \text{g} &: x \mapsto \frac{1}{x - 1} \quad \text{for } x \in \mathbb{R}, \ x \ne 1. \end{aligned}…