Math topic中文

Graph of inverse

Relationship between graphs of a one-to-one function and its inverse.

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. 3A curve with equation y=f(x)y=f(x) is transformed by a reflection in the y-axis, followed by a translation of 1 unit in the…
  4. 4The 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…
  5. 5A curve y=f(x)y=f(x) undergoes in succession, the following transformations: A: A reflection in the y-axis B: A translation…
  6. 6A 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}…
  7. 7The 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…
  8. 8The 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}…
  9. 9The 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}…
  10. 10The 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…
  11. 11The 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}…
  12. 12The 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…
  13. 13A 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}…
  14. 14The 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…
  15. 15The 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)…
  16. 16The 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…
  17. 17The 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}…
  18. 18The 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…
  19. 19The 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,…
  20. 20The 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…
  21. 21The 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…
  22. 22The 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}…
  23. 23The 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,…
  24. 24The 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…
  25. 25The 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…
  26. 26The 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}.…
  27. 27The 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…
  28. 28The 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,…
  29. 29The 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…
  30. 30The 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}…
  31. 31The 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…
  32. 32The function f\text{f} is defined by f(x)=(x+1)∣x+1∣,for x∈R,−4<x≤2.\text{f}(x) = (x+1)|x+1|, \quad \text{for } x \in \mathbb{R}, -4 < x \le 2. Fi…
  33. 33The 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}.…
  34. 34The 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}…
  35. 35It is given that f(x)=a−x21+x2f(x)=\dfrac{a-x^2}{1+x^2} where aa is a real constant and a≠−1a \ne -1. Find the range of values of aa…