$\def\frac{\dfrac}$
$\def\iixi#1#2{\left(\begin{array}{c}#1\\#2\end{array}\right)}\let\frac=\dfrac$
| 1 | (IB/s1/2018/November/Paper2/q3)
[Maximum mark: 7]
Let $f(x)=\frac{6 x-1}{2 x+3}$, for $x \neq-\frac{3}{2}$
| (a) | For the graph of $f$. |
(i) find the $y$-intercept;
(ii) find the equation of the vertical asymptote;
(iii) find the equation of the horizontal asymptote. |
| (b) | Hence or otherwise, write down $\displaystyle\lim _{x \rightarrow \infty}\left(\frac{6 x-1}{2 x+3}\right)$. |
| 2 | (IB/s1/2018/May/paper2tz2/q7)
[Maximum mark: 7 ]
Let $f(x)=\frac{8 x-5}{c x+6}$ for $x \neq-\frac{6}{c}, c \neq 0$
| (a) | The line $x=3$ is a vertical asymptote to the graph of $f$. Find the value of $c .$ |
| (b) | Write down the equation of the horizontal asymptote to the graph of $f$. |
| (c ) | The line $y=k$, where $k \in \mathbb{R}$ intersects the graph of $|f(x)|$ at exactly one point. Find the possible values of $k$. |
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| 3 | (IB/s1/2017/May/paper2tz1/q3)
[Maximum mark: 6]
Consider the graph of $f(x)=\frac{\mathrm{e}^{x}}{5 x-10}+3$, for $x \neq 2$.
| (a) | Find the $y$-intercept. |
| (b) | Find the equation of the vertical asymptote. |
| (c ) | Find the minimum value of $f(x)$ for $x>2$. |
[2]
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| 4 | (IB/sl/2016/May/paper2tz2/q3)
[Maximum mark: 7]
Let $f(x)=e^{a, r}-2$.
| (a) | For the graph of $f$
(i) write down the $y$-intercept:
(ii) find the $x$-intercept:
(iii) write down the equation of the horizontal asymptote. |
| (b) | On the following grid, sketch the graph of $f$, for $-4 \leq x \leq 4$. [3]  |
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| 5 | (IB/s1/2016/May/paper2tz2/q9)
[Maximum mark: 14]
Let $f(x)=\frac{1}{x-1}+2$, for $x>1 .$
| (a) | Write down the equation of the horizontal asymptote of the graph of $f$. |
| (b) | Find $f^{\prime}(x)$. [2] |
Let $g(x)=a \mathrm{e}^{-x}+b$, for $x \geq 1$. The graphs of $f$ and $g$ have the same horizontal asymptote.
| (c) | Write down the value of $b$. |
| (d) | Given that $g^{\prime}(1)=-c$, find the value of $a$. |
| (e) | There is a value of $x$, for $1 < x < 4$, for which the graphs of $f$ and $g$ have the same gradient. Find this gradient. |
|
| 6 | (IB/s1/2018/May/paper1tz1/q3)
[Maximum mark: 7]
Consider a function $f(x)$, for $-2 \leq x \leq 2$. The following diagram shows the graph of $f$.
| (a) | Write down the value of
(i) $f(0):$
(ii) $f^{-1}(1)$. |
| (b) | Write down the range of $f^{-1}$. |
| (c) | On the grid above, sketch the graph of $f^{-1}$. |
|
| 7 | (IB/s1/2019/November/Paper2/q3)
[Maximum mark: 7]
Let $f(x)=x-8, g(x)=x^{4}-3$ and $h(x)=f(g(x))$
Let $\mathrm{C}$ be a point on the graph of $h$. The tangent to the graph of $h$ at $\mathrm{C}$ is parallel to the graph of $f$.
| (b) | Find the $x$-coordinate of $\mathrm{C}$. |
|
| 8 | (IB/s1/2019/May/paper1tz1/q4)
[Maximum mark: 6]
Let $f(x)=\frac{2 x-1}{x+3}, x \neq-3$.
| (a) | Write down the equation of the vertical asymptote of the graph of $f$. [1] |
| (b) | Find $f^{-1}(x)$. $[3]$ |
| (c) | Find the equation of the horizontal asymptote of the graph of $f^{-1}$. [2] |
|
| 9 | (IB/s1/2019/May/paper1tz2/q3)
[Maximum mark: 6]
Consider the function $f(x)=\frac{3 x+1}{x-2}, x \neq 2$.
| (a) | For the graph of $f$.
(i) write down the equation of the vertical asymptote;
(ii) find the equation of the horizontal asymptote. |
Let $g(x)=x^{2}+4, x \in \mathbb{R}$,
| (b) | Find $(f \circ g)(1)$ |
|
| 10 | (IB/s1/2019/May/paper2tz1/q8)
[Maximum mark: 13]
Let $f(x)=2 \sin (3 x)+4$ for $x \in \mathbb{R}$.
| (a) | The range of $f$ is $k \leq f(x) \leq m .$ Find $k$ and $m$. |
Let $g(x)=5 f(2 x)$.
| (b) | Find the range of $g$. |
The function $g$ can be written in the form $g(x)=10 \sin (b x)+c$.
| (c) | (i) Find the value of $b$ and of $c$.
(ii) Find the period of $g$. |
| (d) | The equation $g(x)=12$ has two solutions where $\pi \leq x \leq \frac{4 \pi}{3}$. Find both solutions: [3] |
|
| 11 | (IB/s1/2019/May/paper2tz1/q9)
[Maximum mark: 16]
Let $f(x)=\frac{16}{x}$. The line $L$ is tangent to the graph of $f$ at $x=8$.
| (a) | Find the gradient of $L$. [2] |
$L$ can be expressed in the form $r=\left(\begin{array}{l}8 \\ 2\end{array}\right)+i u$.
The direction vector of $y=x$ is $\left(\begin{array}{l}1 \\ 1\end{array}\right)$.
| (c) | Find the acute angle between $y=x$ and $L$. |
| (d) | (i) Find $(f \circ f)(x)$.
(ii) Hence, write down $f^{-1}(x)$.
(iii) Hence or otherwise, find the obtuse angle formed by the tangent line to $f$ at $x=8$ and the tangent line to $f$ at $x=2$.
[7] |
|
| 12 | (IB/sl/2018/November/Paper1/q2)
[Maximum mark: 5]
Two functions, $f$ and $g$, are defined in the following table. $$\begin{array}{|c|r|r|r|r|}\hline x & -2 & 1 & 3 & 6 \\\hline f(x) & 6 & 3 & 1 & -2 \\\hline g(x) & -7 & -2 & 5 & 9 \\\hline\end{array}$$
| (a) | Write down the value of $f(1)$. |
| (b) | Find the value of $(g \circ f)(1)$. |
| (c) | Find the value of $g^{-1}(-2)$. [2] |
|
| 13 | (IB/s1/2018/May/paper1tz1/q1)
[Maximum mark: 6]
Let $f(x)=\sqrt{x+2}$ for $x \geq-2$ and $g(x)=3 x-7$ for $x \in \mathbb{R}$.
| (b) | Find $(g \circ f)(14)$. |
|
| 14 | (IB/s1/2018/May/paper1tz1/q7)
[Maximum mark: 7 ]
Consider $f(x), g(x)$ and $h(x)$, for $x \in \mathbb{R}$ where $h(x)=(f \circ g)(x)$.
Given that $g(3)=7, g^{\prime}(3)=4$ and $f^{\prime}(7)=-5$, find the gradient of the normal to the curve of $h$ at $x=3$.
|
| 15 | (IB/sl/2017/November/Paper 1/q3)
[Maximum mark: 6]
The following diagram shows the graph of a function $f$, with domain $-2 \leq x \leq 4$.
The points $(-2,0)$ and $(4,7)$ lie on the graph of $f$.
| (a) | Write down the range of $f$. |
| (b) | Write down
(i) $f(2)$;
(ii) $f^{-1}(2)$. [2] |
| (c) | On the grid opposite, sketch the graph of $f^{-1}$. $[3]$ |
|
| 16 | (IB/s1/2017/November/Paper1/q5)
[Maximum mark: 6]
Let $f(x)=1+\mathrm{e}^{-x}$ and $g(x)=2 x+b$, for $x \in \mathbb{R}$, where $b$ is a constant.
| (a) | Find $(g \circ f)(x)$. [2] |
| (b) | Given that $\lim _{x \rightarrow+\infty}(g \circ f)(x)=-3$, find the value of $b$. $[4]$ |
|
| 17 | (IB/s1/2017/May/paper1tz1/q2)
[Maximum mark: 5]
Let $f(x)=5 x$ and $g(x)=x^{2}+1$, for $x \in \mathbb{R}$.
| (b) | Find $(f \circ g)(7)$. |
|
| 18 | (IB/s1/2017/May/paper1tz2/q6)
[Maximum mark: 5]
The values of the functions $f$ and $g$ and their derivatives for $x=1$ and $x=8$ are shown in the following table, $$\begin{array}{|c|c|c|c|c|}\hline x & f(x) & f^{\prime}(x) & g(x) & g^{\prime}(x) \\\hline 1 & 2 & 4 & 9 & -3 \\\hline 8 & 4 & -3 & 2 & 5 \\\hline\end{array}$$
Let $h(x)=f(x) g(x)$.
| (b) | Find $h^{\prime}(8)$. [3] |
|
| 19 | (IB/s1/2017/May/paper2tz2/q3)
[Maximum mark: 6]
The following diagram shows the graph of a function $y=f(x)$, for $-6 \leq x \leq-2$.
The points $(-6,6)$ and $(-2,6)$ lie on the graph of $f$. There is a minimum point at $(-4,0)$.
| (a) | Write down the range of $f$. |
Let $g(x)=f(x-5)$
| (b) | On the grid above, sketch the graph of $g$. [2] |
| (c) | Write down the domain of $g$. [2] |
|
| 20 | (IB/s1/2016/May/paper1tz1/q1)
[Maximum mark: 5]
Let $f(x)=8 x+3$ and $g(x)=4 x$, for $x \in \mathbb{R}$
| (a) | Write down $g(2)$. [1] |
| (b) | Find $(f \circ g)(x)$ [2] |
| (c ) | Find $f^{-1}(x)$. [2] |
|
| 21 | (IB/s1/2016/May/paper1tz2/q6)
[Maximum mark: 7]
Let $f(x)-6 x \sqrt{1-x^{2}}$, for $-1 \leq x \leq 1$, and $g(x)-\cos (x)$, for $0 \leq x \leq \pi$
Let $h(x)=(f \circ g)(x)$
| (a) | Write $h(x)$ in the form $a \sin (b x)$, where $a, b \in \mathbb{Z}$. |
| (b) | Hence find the range of $h$. [2] |
|
| 22 | (IB/s1/2016/May/paper2tz1/q2)
[Maximum mark: 6]
Let $f(x)=x^{2}$ and $g(x)=3 \ln (x+1)$, for $x>-1$
| (a) | Solve $f(x)=g(x)$.[3] |
| (b) | Find the area of the region enclosed by the graphs of $f$ and $g$. [3] |
|
| 23 | (IB/sl/2015/May/paper1tz1/q4)
[Maximum mark: 7]
The following diagram shows the graph of a function $f$.
| (b) | Find $(f \circ f)(-1)$. |
| (c) | On the same diagram, sketch the graph of $y=f(-x)$. |
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| 24 | (IB/sl/2015/May/paper1tz2/q6)
[Maximum mark: 8]
Let $f(x)=a x^{3}+b x .$ At $x=0$, the gradient of the curve of $f$ is 3 . Given that $f^{-1}(7)=1$, find the value of $a$ and of $b$.
|
| 25 | (IB/sl/2015/May/paper2tz1/q4)
[Maximum mark: 7]
Let $f(x)=\frac{2 x-6}{1-x}$, for $x \neq 1$
| (a) | For the graph of $f$
(i) find the $x$-intercept;
(ii) write down the equation of the vertical asymptote;
(iii) find the equation of the horizontal asymptote. |
| (b) | Find $\lim _{x \rightarrow \infty} f(x)$. |
|
| 26 | (IB/sl/2015/November/Paper1/q4)
[Maximum mark: 7 ]
Let $f(x)=3 \sin (\pi x)$.
| (a) | Write down the amplitude of $f$. |
| (b) | Find the period of $f$. |
| (c) | On the following grid, sketch the graph of $y=f(x)$, for $0 \leq x \leq 3$. $[4]$ |
|
| 27 | (IB/sl/2015/November/Paper1/q5)
[Maximum mark: 6]
Let $f(x)=(x-5)^{3}$, for $x \in \mathbb{R}$
| (b) | Let $g$ be a function so that $(f \circ g)(x)=8 x^{6}$. Find $g(x)$. [3] |
|
| 28 | (IB/sl/2015/November/Paper2/q3)
[Maximum mark: 7]
Let $f(x)=2 \ln (x-3)$, for $x>3 .$ The following diagram shows part of the graph of $f$
| (a) | Find the equation of the vertical asymptote to the graph of $f$. |
| (b) | Find the $x$-intercept of the graph of $f$. |
| (c) | The region enclosed by the graph of $f$, the $x$-axis and the line $x=10$ is rotated $360^{\circ}$ about the $x$-axis. Find the volume of the solid formed. [3] |
|
Answer
1 (a)(i) $\quad\left(0,-\frac{1}{3}\right)$ (ii) $x=-\frac{3}{2}$ (iii) $y=3$ (b) 3
2 (a) $c=-2$ (b) $y=-4$ (c) $k=4, k=0$
3 (a) $y$-intercept is $2.9$ (b) $x=2$ (c) $7.02$
4 (a) (i) $y=-1$ (ii) $x=2 \ln 2$ (iii) $y=-2$ (b) Graph
5 (a) $y=2$ (b) $f^{\prime}(x)=\frac{-1}{(x-1)^{2}}$ (c) $b=2$ (d) $a=e^{2}$ (e) $-1$
6 (a)(i) $f(0)=-\frac{1}{2}$ (ii) $f^{-1}(1)=2$ (b) $-2 \leqslant y \leqslant 2$ (c) Graph
7 (a) $h(x)=x^4-11$ (b) $x=\sqrt[3]{\frac{1}{4}}$
8 (a) $x=-3$ (b) $f^{-1}(x)=\frac{-3x-1}{x-2}$ (c) $y=-3$
9 (a) (i)$ x=2 $ (ii)$ y=3 $ (b) $\frac{16}{3}$
10 (a) $k=2,m=6$ (b) $10\le y\le 30$ (c)(i) $b=6,c=20$ (ii) $\frac{\pi}{3}$ (d) $3.82,4.03$
11 (a) $-0.25$ (b) $\iixi{4}{-1}$ (c) 1.03 (d) (i) $(f\circ f)(x)=x$(ii) $f^{-1}(x)=\frac{16}{x}$ (iii) 2.06 or $118^{\circ}$
12 (a) $f(1)=3$ (b) $(g \circ f)(1)=5$ (c) $g^{-1}(-2)=1$
13 (a) $\quad f(14)=4$ (b) 5 (c) $g^{-1}(x)=\frac{x+7}{3}$
14 $\frac{1}{20}$
15 (a) $0 \leqslant y \leqslant 7$ (b) (i) $f(2)=3$ (ii) $f^{-1}(2)=0$ (c) Graph
16 (a) $(g \circ f)(x)=2\left(1+e^{-x}\right)+b$ (b) $b=-5$
17 (a) $f^{\prime}(x)=\frac{x}{5}$ (b) $(f \circ g)(7)=250$
18 (a) $h(1)=18$ (b) $h^{\prime}(8)=14$
19 (a) $0 \leqslant y \leqslant 6$ (b) Graph (c) $-1 \leqslant x \leqslant 3$
20 (a) 8 (b) $(f\circ g)(x)=32x+3$ (c) $f^{-1}(x)=\frac{x-3}{8}$
21 (a) $h(x)=3 \sin (2 x)$ (b) $-3 \leqslant y \leqslant 3$
22 (a) $\quad x=0, x=1.74$ (b) $1.31$
23(a) $\bar{f}^{\prime}(-1)=5$ (b) $(f\circ f )(-1)=1$ (c) Graph
24(a) $a=4, b=3$
25(a) (i) $(3,0)$ (ii) $x=1$ (iii) $y=-2$ (b) $-2$
26(a) $3$ (b) 2 (c) Graph
27(a) $\quad f^{-1}(x)=\sqrt[3]{x}+5$ (b) $g(x)=2 x^{2}+5$
28(a) $x=3$ (b) $x=4$ (c) volume $=142$
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