Math topic中文

Concepts

Concepts of function, domain and range.

Practice questions

1The functions $f$ and $g$ are defined by $f : x \mapsto \sqrt{16 - 4x},\; x \in \mathbb{R},\; x \le 4,$ $g : x \mapsto x^{2},\; x \in \mathbb{R}.$ …
2Show that $\cos 3\theta = 4\cos^{3}\theta - 3\cos\theta$ . Hence, or otherwise, evaluate $\displaystyle \int_{0}^{\pi/6} \sin 2\theta\,\cos 3\theta\, d\theta$ …
3函数 $f\left(x\right)=\begin{cases}{2}^{x}-2\;,\;x\leqslant 1\;,\;\\ {\log}_{2}\left(x-1\right)\;,\;x>1\end{cases}\;$ 则 $f\left[f\left(\dfrac{5}{2}\right)\right]=$ …
4A curve with equation $y=f(x)$ is transformed by a reflection in the y-axis, followed by a translation of 1 unit in the…
5The function $\text{g}$ is defined by $\text{g} : x \to \sin x + x$ , for $x \in \mathbb{R}$ , $-\pi \le x \le \pi$ . Show…
6The diagram shows the curve $y = \text{f}(1-x)$ . The curve has a maximum point at $(-1, 5)$ and two asymptotes $x = -2$ …
7The function $g$ is defined by $g : x \mapsto \ln(x - \sqrt{3})$ , for $x \in \mathbb{R}$ , $x > \sqrt{3}$ . Explain how yo…
8A curve $y=f(x)$ undergoes in succession, the following transformations: A: A reflection in the y-axis B: A translation…
9It is given that $y = \sin\left(e^x - 1\right)$ . Show that $\dfrac{d^2 y}{dx^2} = \dfrac{dy}{dx} - ye^{2x}$ . Hence, find…
10The function $f$ is defined by $f : x \mapsto x^2 - 4x,\ x \in \mathbb{R},\ |x|<1.$ The function $g$ is defined by $g : x \mapsto 1 + \frac{1}{x+3},\ x \in \mathbb{R},\ x \ne -3.$ …
11The function f, with domain the set of all real values, is given by $\text{f}(x) = \begin{cases} 4 - x^2 & \text{for } 0 < x \le 2, \\ 2x - 4 & \text{for } 2 < x \le 4, \end{cases}$ …
12Section A: Pure Mathematics [60 marks] The curve $C$ has equation $y = \frac{x^2 - 2x + 4}{x - b}$ , $x \ne b$ . Given tha…
13The functions f, g and h are defined as follows: f : $x \rightarrow \sin(\pi x), x \in \mathbb{R}, 0 \le x \le 2$ , g : $x \rightarrow a^{x}, x \in \mathbb{R}, x \ge -1$ …
14函数 $y= \sqrt{{x}^{2}-2x+3}+{\log }_{3}(x+2)$ 的定义域为( ) A. $(-,∞-1)∪(3,+∞)$ B. $(-∞,-1)∪[3,+∞)$ C. $(-2,-1]$ D.…
15Section A: Pure Mathematics 1 (a) Show that $\dfrac{1}{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}=\dfrac{1}{2x^{2}}\big(\sqrt{1+x^{2}}+\sqrt{1-x^{2}}\big).$ …
16已知函数 $f\left(x\right)$ 满足 $f\left(x+1\right)={{x}^{2}}-2$ ，则 $f\left(3\right)=(\quad)$ A. $2$ B. $4$ C. $7$ D. $14$ …
17下列各组中的函数 $f(x)$ 与 $g(x)$ 相等的是 $(\quad)$ A. $f(x)=|x|$ ， $g(x)=(\sqrt{x})^{2}$ B. $f(x)=\dfrac{x^{2}-1}{x+1}$ ， $g(x)=x-1$ …
18The functions f and g are defined by $f: x \mapsto |(x+1)(x-3)|,\quad x \in \mathbb{R},$ $g: x \mapsto x-2,\quad x \in \mathbb{R}\text{ and }x \ge -1.$ …
19若 $\dfrac{\sqrt{x+2}}{x-2}$ 在实数范围内有意义，则 $x$ 的取值范围是 ___ .
20Functions f and g are defined by $f : x \mapsto [\ln(x-1)]^{2} + 2,\quad x \ge a,$ $g : x \mapsto 4 + 3x - x^{2},\quad x \le \frac{3}{2}.$ …
21The function f is defined by $f: x \mapsto \dfrac{1}{1 - x^2}$ , $-1 < x \le 0$ . (i) Define $f^{-1}$ . (ii) Sketch the gra…
22A function g is defined by $g(x)=\begin{cases} \displaystyle\frac{2}{1+x^2} &\text{if } x\le 0,\\[4pt] 0 &\text{if } x>0. \end{cases}$ …
23The function $\mathrm{f}$ is defined by $\mathrm{f} : x \mapsto \sqrt{5+2x}$ , for $x \in \mathbb{R}$ , $x > -\frac{5}{2}$ …
24The functions f and g are defined by $f : x \mapsto x^2 + 3x + 1,\ x \in \mathbb{R},\ x \le -\frac{3}{2},$ $g : x \mapsto 3 + e^{-x},\ x \in \mathbb{R}.$ …
25The diagram shows the graph of $y = f(x)$ with asymptotes $y = k$ , $x = 0$ and the graph cuts the x-axis at $(a,0)$ . On…
26It is given that $f(x) = \begin{cases} \tfrac{1}{2}x^2, & \text{for } 1 \le x < 2, \\ \tfrac{1}{4}(-3x+14), & \text{for } 2 \le x < 4, \end{cases}$ …
27A curve $C$ is defined parametrically by $x = a \tan t, \quad y = a \sec^2 t \sin t, \quad -\frac{\pi}{4} < t < \frac{\pi}{4},$ …
28The function f is defined by $f : x \mapsto e^x + \dfrac{1}{2x+2}\quad\text{for } x \in \mathbb{R},\ x \ne -1.$ A func…
29A function $\text{f}$ is defined by $\text{f}(x) = x^3 + ax^2 + bx + c$ , where $a$ , $b$ and $c$ are constants. The graph…
30代数式 $\sqrt{2-x}+\dfrac{1}{x-3}$ 中自变量 $x$ 的取值范围是 $(\quad)$ A. $x\leqslant 2$ B. $x=3$ C. $x< 2$ 且 $x≠3$ D. $x\leqslant 2$ …
31已知函数 $f(x)=a^{x}+b(a>0$ 且 $a\neq 1)$ 的图像过点 $(1,7)$ ，其反函数 $y=f^{-1}(x)$ 的图像过点 $(4,0)$ ，则 $a$ 的值为___.
32已知函数 $f\left(x\right)$ 是定义在 $R$ 上的偶函数，且 $f\left(0\right)=0$ ，当 $x>0$ 时， $f\left(x\right)={{\log}_{2}}\left(x+1\right).(1)$ …
33The function $\mathrm{f}$ is defined by $\mathrm{f}(x) = \begin{cases} 4 - x & \text{for } 1 \le x < 3, \\ (x - 4)^2 & \text{for } 3 \le x < 4, \end{cases}$ …
34It is given that $f(x)=\begin{cases}4x,&0\le x<1\\(3-x)^2,&1\le x\le 3\end{cases}$ and $f(-x)=-f(x)$ . It is also kno…
35The function f is defined by $f : x \mapsto \dfrac{1}{x^2 + 6x + 5}$ , for $x \ge -3$ , $x \ne -1$ . The function g is defi…
36设函数 $f(x)$ 的定义域为 $R$ ，满足 $f(x+1)=2f(x)$ ，且当 $x\in(0,1]$ 时， $f(x)=x^{2}-x$ ，则 $f(\dfrac{7}{2})$ 的值为 ___.
37It is given that $y = \ln(2 - \text{e}^{-2x})$ . Show that $\frac{\text{d}y}{\text{d}x} = 4\text{e}^{-y} - 2$ . Hence find…
38Topic 4: Functions [MI/2014/MYE PU1/13] The function $\mathrm{f}$ is defined by $\mathrm{f} : x \mapsto x^2 - 6x + 14, \quad x \in \mathbb{R}.$ …
39In a triangle with vertices A, B and C, angle $BAC$ is a right-angle and angle $ABC=\frac{\pi}{3}-x$ . Show that $\dfrac{AB}{AC}=\dfrac{1+\sqrt{3}\tan x}{\sqrt{3}-\tan x}$ …
40Sketch the curve with equation $y = \ln(x+1), x > -1$ , stating the equation of asymptote and intercepts with the axes. O…
41The function $\text{f}$ is defined by $\text{f} : x \mapsto \frac{1}{x^2 - x - 6} + 2, \quad x \in \mathbb{R}, \ x \ne -2, \ x \ne 3.$ …
42The function $\text{f}$ is defined by $\text{f} : x \to 2x - \frac{1}{2x}, \quad 0 < x < 2.$ It is given that $\text{f}^{-1}$ …
43The equation of a curve $y = \mathrm{f}(x)$ is in the form of a cubic polynomial. The curve has a maximum point at $(-1, 0)$ …
44Given that $\mathrm{f}(x) = 1 + \cos 2x$ , find $\mathrm{f}(0)$ , $\mathrm{f}'(0)$ , $\mathrm{f}''(0)$ , $\mathrm{f}^{(3)}(0)$ …
45函数 $y=\dfrac{ \sqrt{x+1}}{x-3}$ 的自变量 $x$ 的取值范围是___．
46The curves $C_{1}$ and $C_{2}$ have equations $\dfrac{x^{2}}{9} - \dfrac{y^{2}}{4} = 1$ and $y^{2} + x^{2} = k^{2}$ resp…
47Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f} : x \to \frac{4}{(x-4)^2}, \quad x \in \mathbb{R}, x \ne 4,$ …
48The function f is defined by $f : x \mapsto \dfrac{5}{(x-4)^2}$ for $x\in\mathbb{R},\ x<4$ . Find $f^{-1}(x)$ and state t…
49下列函数的定义域判断正确的是 $(\quad)$ A. $f(x)=\dfrac{3x}{x-4}$ 的定义域为 $(-∞,0)∪(0,+∞)$ B. $f(x)=\sqrt{x^{2}}$ 的定义域为 $[0,+∞)$ C. $f(x)=\dfrac{6}{x^{2}-3x+2}$ …
50Find the set of values of $x$ such that $-1<\dfrac{2x+3}{x-1}<1$ . Deduce the range of values of $x$ such that $-1<\dfrac{2x+1}{x-2}<1$ …
51定义： $[x]$ 表示不大于 $x$ 的最大整数，例如： $[2.3]=2$ ， $[1]=1.$ 以下结论：① $[-1.21]=-2$ ；② $[a-1]=[a]-1$ ；③当 $-1 < x < 1$ 时， $[1+x]+[1-x]$ …
52已知函数f（x）的定义域为[0，2]，则f（2x）的定义域为．
53The function f is defined by $f : x \mapsto 4 + \dfrac{1}{(x-1)^2 + 1},\quad x \ge 0$ . The function g is defined by g :…
54Show that $\dfrac{d}{dx}(\tan x)=\sec^2 x$ Show that $\sin 2x\cot x = 2\cos^2 x$ Hence, find the exact value of $\displaystyle\int_{\tfrac{\pi}{30}}^{\tfrac{\pi}{5}} \csc(10x)\tan(5x)\,dx$ …
55已知函数f（x）满足f（x）+2f（-x）=3x，则f（1）=___．
56The function $\text{g}$ is defined by $\text{g} : x \mapsto \ln(x - \sqrt{3})$ , for $x \in \mathbb{R}$ , $x > \sqrt{3}$ .…
57The function $\mathrm{f}$ is defined by $\mathrm{f} : x \mapsto \frac{ax}{x-a}, \quad \text{for } x \in \mathbb{R}, \ -a \le x < a,$ …
58The functions f and g are defined by $f : x \mapsto 1-\dfrac{1}{x^{2}-1}$ for $x \in \mathbb{R},\; x<-1$ , $g : x \mapsto x^{2}+4x+5$ …
59A curve has parametric equations $x = 2t - \frac{1}{t}$ and $y = t + \frac{1}{t}$ where $t$ is a parameter and $t \ne 0$ …
60Complete the square for $2x^2 - 9x + 5$ . In the rest of the question, the domain of $\mathrm{f}$ is $x \in \mathbb{R}$ ,…