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Date	May 2017	Marks available	1	Reference code	17M.2.AHL.TZ1.H_12
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Explain	Question number	H_12	Adapted from	N/A

Question

Consider $f(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right)$

The function $f$ is defined by $f(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right),{\text{ }}x \in D$

The function $g$ is defined by $g(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right),{\text{ }}x \in \left] {1,{\text{ }}\infty } \right[$ .

Find the largest possible domain $D$ for $f$ to be a function.

[2]

Sketch the graph of $y = f(x)$ showing clearly the equations of asymptotes and the coordinates of any intercepts with the axes.

[3]

Explain why $f$ is an even function.

[1]

Explain why the inverse function ${f^{ - 1}}$ does not exist.

[1]

Find the inverse function ${g^{ - 1}}$ and state its domain.

[4]

Find $g'(x)$ .

[3]

Hence, show that there are no solutions to $g'(x) = 0$ ;

[2]

g.i.

Hence, show that there are no solutions to $({g^{ - 1}})'(x) = 0$ .

[2]

g.ii.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

${x^2} - 1 > 0$ (M1)

$x < - 1$ or $x > 1$ A1

[2 marks]

M17/5/MATHL/HP2/ENG/TZ1/12.b/M

shape A1

$x = 1$ and $x = - 1$ A1

$x$ -intercepts A1

[3 marks]

EITHER

$f$ is symmetrical about the $y$ -axis R1

$f( - x) = f(x)$ R1

[1 mark]

EITHER

$f$ is not one-to-one function R1

horizontal line cuts twice R1

Note: Accept any equivalent correct statement.

[1 mark]

$x = - 1 + \ln \left( {\sqrt {{y^2} - 1} } \right)$ M1

${{\text{e}}^{2x + 2}} = {y^2} - 1$ M1

${g^{ - 1}}(x) = \sqrt {{{\text{e}}^{2x + 2}} + 1} ,{\text{ }}x \in \mathbb{R}$ A1A1

[4 marks]

$g'(x) = \frac{1}{{\sqrt {{x^2} - 1} }} \times \frac{{2x}}{{2\sqrt {{x^2} - 1} }}$ M1A1

$g'(x) = \frac{x}{{{x^2} - 1}}$ A1

[3 marks]

$g'(x) = \frac{x}{{{x^2} - 1}} = 0 \Rightarrow x = 0$ M1

which is not in the domain of $g$ (hence no solutions to $g'(x) = 0$ ) R1

[2 marks]

g.i.

$({g^{ - 1}})'(x) = \frac{{{{\text{e}}^{2x + 2}}}}{{\sqrt {{{\text{e}}^{2x + 2}} + 1} }}$ M1

as ${{\text{e}}^{2x + 2}} > 0 \Rightarrow ({g^{ - 1}})'(x) > 0$ so no solutions to $({g^{ - 1}})'(x) = 0$ R1

Note: Accept: equation ${{\text{e}}^{2x + 2}} = 0$ has no solutions.

[2 marks]

g.ii.

Examiners report

[N/A]

g.i.

[N/A]

g.ii.

Syllabus sections

Topic 2—Functions » SL 2.2—Functions, notation domain, range and inverse as reflection

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18M.2.SL.TZ2.T_6b:
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18M.2.SL.TZ1.T_4b:
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18M.1.AHL.TZ2.H_10b.i:
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20N.1.SL.TZ0.T_2a.ii:
State, in the context of the question, what the value of $8.50$ represents.
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Find the maximum number of large cheese pizzas that Kaelani can order from Olava’s Pizza Company.
17N.2.AHL.TZ0.H_10a.ii:
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17M.1.AHL.TZ1.H_11c:
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20N.1.SL.TZ0.T_2b:
Write down the minimum number of pizzas that can be ordered.
17M.1.SL.TZ1.T_12a:
Write down the domain of the function.
16N.2.AHL.TZ0.H_5c:
Solve the inequality $\left| {3x\arccos (x)} \right| > 1$ .
17M.2.AHL.TZ1.H_12f:
Find $g'(x)$ .
16N.2.SL.TZ0.S_1a:
Find $f(8)$ .
17M.2.SL.TZ2.S_6c:
The equation $(f \circ g)(x) = k$ has exactly two solutions, for $0 \leqslant x \leqslant 2.25$ . Find the possible values of $k$ .
EXN.2.SL.TZ0.3c:
Find the range of $h$ .
18M.2.SL.TZ2.T_6d:
Find $\frac{{{\text{dy}}}}{{{\text{dx}}}}$ .
18M.1.AHL.TZ1.H_9b.ii:
The coordinates of B can be expressed in the form B $\left( {{2^a},\,b \times {2^{ - 3a}}} \right)$ where a, b $\in \mathbb{Q}$ . Find the value of a and the value of b.
17M.2.AHL.TZ1.H_12g.i:
Hence, show that there are no solutions to $g'(x) = 0$ ;
17N.1.AHL.TZ0.H_6b:
Hence or otherwise, solve the inequality $\left| {\frac{{1 - 3x}}{{x - 2}}} \right| < 2$ .
21M.1.SL.TZ1.4a:
Find the total cost of buying $40$ litres of gas at Leon’s gas station.
18M.2.SL.TZ1.T_4c:
Use your answer to part (b) to show that the minimum value of f(x) is −22 .
21M.1.SL.TZ1.4b:
Find $L^{- 1} (70)$ .
18M.2.AHL.TZ2.H_10d.i:
Find $\alpha$ and β in terms of $k$ .
19M.2.SL.TZ2.T_5d:
Find $f'\left( x \right)$ .
17M.1.SL.TZ1.T_12b.i:
Draw the line $y = - 6$ on the axes.
19M.2.SL.TZ1.T_4d:
Find the distance from the centre of Orangeton to the point at which the road meets the highway.
19M.1.AHL.TZ1.H_8b:
find the value of $f\left( 1 \right)$ .
19N.1.SL.TZ0.T_10b:
While in Kota Kinabalu, Criselda spent $440 MYR$ . She returned to the Currency Exchange counter and changed the remainder of her $MYR$ into $USD$ .

Calculate the amount of $USD$ she received.
17M.2.SL.TZ2.S_6a:
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19M.2.AHL.TZ2.H_4b:
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17N.1.SL.TZ0.S_5b:
Given that $\mathop {\lim }\limits_{x \to + \infty } (g \circ f)(x) = - 3$ , find the value of $b$ .
16N.2.SL.TZ0.S_1c:
Solve $(g \circ f)(x) = 0$ .
18M.1.AHL.TZ2.H_10a:
Find the inverse function ${f^{ - 1}}$ , stating its domain.
19M.2.SL.TZ1.S_9a:
Find the gradient of $L$ .
19N.1.SL.TZ0.T_10a:
Calculate the amount of $MYR$ that Criselda received.
19M.2.SL.TZ1.S_9d.ii:
Hence, write down ${f^{ - 1}}\left( x \right)$ .
18M.2.AHL.TZ2.H_10a.iv:
Explain why $f$ is not a function for $- \frac{{3\pi }}{4} \leqslant x \leqslant \frac{\pi }{4}$ .
16N.2.AHL.TZ0.H_5a:
Sketch the graph of $f$ indicating clearly any intercepts with the axes and the coordinates of any local maximum or minimum points.
17M.1.SL.TZ1.T_12c:
Find the range of values of $k$ for which $f(x) = k$ has no solution.
17M.2.AHL.TZ1.H_12b:
Sketch the graph of $y = f(x)$ showing clearly the equations of asymptotes and the coordinates of any intercepts with the axes.
17M.1.SL.TZ1.T_12b.ii:
Write down the number of solutions to $f(x) = - 6$ .
21M.1.SL.TZ2.7b:
Find the value of $f^{- 1} (0)$ .
21M.1.SL.TZ2.7a:
Find the range of $f$ .
21M.1.AHL.TZ2.2c:
Write down the range of $f^{- 1} (x)$ .
17M.2.SL.TZ2.S_3a:
Write down the range of $f$ .
17M.2.AHL.TZ1.H_12a:
Find the largest possible domain $D$ for $f$ to be a function.
19M.1.AHL.TZ1.H_8c:
find the value of $f\left( 4 \right)$ .
21M.1.AHL.TZ2.2b:
Find an expression for the inverse function $f^{- 1} (x)$ . The domain is not required.
21M.1.AHL.TZ2.2a:
Find the range of $f$ .
18M.2.AHL.TZ2.H_10b:
Show that $g\left( t \right) = {\left( {\frac{{1 + t}}{{1 - t}}} \right)^2}$ .
17M.1.AHL.TZ1.H_11f:
Determine the area of the region enclosed between the graph of $y = f\left( {\left| x \right|} \right)$ , the $x$ -axis and the lines with equations $x = - 1$ and $x = 1$ .
17M.1.AHL.TZ1.H_11b:
Sketch the graph of $f(x)$ , indicating on it the equations of the asymptotes, the coordinates of the $y$ -intercept and the local maximum.
19M.2.SL.TZ2.T_5e:
Find the gradient of the graph of $y = f\left( x \right)$ at $x = 2$ .
EXN.2.SL.TZ0.3b.i:
Find the value of $t$ at which the ball hits the ground.
EXN.2.SL.TZ0.3a:
Find an expression for the height $h$ of the ball at time $t$ .
21N.1.SL.TZ0.5b.i:
Find $h^{- 1} (10)$ .
21N.1.SL.TZ0.5b.ii:
In the context of the question, interpret your answer to part (b)(i).
21N.1.SL.TZ0.5b.iii:
Write down the range of $h^{- 1}$ .

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