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Date	November 2021	Marks available	2	Reference code	21N.1.SL.TZ0.5
Level	Standard Level	Paper	Paper 1	Time zone	Time zone 0
Command term	Find	Question number	5	Adapted from	N/A

Question

Let the function $h (x)$ represent the height in centimetres of a cylindrical tin can with diameter $x cm$ .

$h (x) = \frac{640}{x^{2}} + 0.5$ for $4 \leq x \leq 14$ .

The function $h^{- 1}$ is the inverse function of $h$ .

Find the range of $h$ .

[3]

Find $h^{- 1} (10)$ .

[2]

b.i.

In the context of the question, interpret your answer to part (b)(i).

[1]

b.ii.

Write down the range of $h^{- 1}$ .

[1]

b.iii.

Markscheme

$h (4) = \frac{640}{4^{2}} + 0.5$ OR $h (14) = \frac{640}{14^{2}} + 0.5$ (M1)

Note: Award (M1) for substituting $4$ or $14$ into $h$ . This can be implicit from seeing $3.77 (3.76530 \dots)$ or $40.5$ .

$3.77 \leq h (x) \leq 40.5 (3.76530 \dots \leq h (x) \leq 40.5)$ A1A1

Note: Award A1 for both correct endpoints seen, A1 for the endpoints in a correct interval.

[3 marks]

$h (x) = 10$ OR $h^{- 1} (x) = \sqrt{\frac{640}{x - 0.5}}$ OR $h^{- 1} (10) = \sqrt{\frac{640}{10 - 0.5}}$ (M1)

$(x =) 8.21 cm (8.20782 \dots)$ A1

[2 marks]

b.i.

a tin that is $10 cm$ high will have a diameter of $8.21 cm (8.20782 . . .)$ A1

Note: Condone a correct answer expressed as the converse.

[1 mark]

b.ii.

$4 \leq h^{- 1} \leq 14$ A1

Note: Accept $4 \leq y \leq 14$ . Do not FT in this part.

[1 mark]

b.iii.

Examiners report

Part (a) was reasonably well done. Many candidates were able to find the endpoints but there was some confusion about whether to use strict or weak inequalities. Some candidates wrote their answer as $40.5 \geq y \geq 3.77$ while some others wrote $40.5 \leq y \leq 3.77$ . A few candidates used integer $x$ values from $4$ to $14$ to find corresponding values for $h (x)$ and gave the full list as their final answer. In part (b), the most popular incorrect answer seen was $6.9$ with weaker candidates simply finding $h (10)$ . Several candidates equated $h (x)$ to $10$ but missed out $+ 0.5$ in their equation. Finding a value of the inverse of a function still proves to be difficult for candidates. There were many candidates who attempted to find an expression for the inverse before substituting $x$ by $10$ and this proved to be difficult for this function. Regardless of what answer candidates derived for part (b), very few of them could write an interpretation of their answer in context. There was significant confusion between the value for the height and value for the diameter. In part (d), there were very few candidates who realized the relationship between the domain of the function and the range of the inverse function. Many candidates simply reverted to their answer to part (a).

b.i.

b.ii.

b.iii.

Syllabus sections

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

Show 123 related questions

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18M.1.AHL.TZ2.H_2a:
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18N.1.AHL.TZ0.H_3a:
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17M.1.AHL.TZ1.H_6b:
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19M.2.SL.TZ1.S_9b:
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19M.2.SL.TZ2.T_5f:
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18M.2.SL.TZ1.T_4a:
Find the value of k.
17M.1.SL.TZ2.T_14a:
Write down the amount of money Jashanti saves per month.
18N.1.AHL.TZ0.H_3c.i:
For the value of $a$ found in part (b), write down the domain of ${g^{ - 1}}$ .
19M.2.SL.TZ1.T_4b.ii:
State the domain of $P$ .
18M.2.SL.TZ2.T_6f:
Given that y = 2x³ − 9x² + 12x + 2 = k has three solutions, find the possible values of k.
17N.1.SL.TZ0.S_5a:
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19M.2.SL.TZ1.S_9d.iii:
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18M.2.SL.TZ2.T_6a:
Sketch the curve for −1 < x < 3 and −2 < y < 12.
19M.1.AHL.TZ1.H_8d:
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18M.2.AHL.TZ2.H_10a.ii:
With reference to your graph, explain why $f$ is a function on the given domain.
17M.2.AHL.TZ1.H_12e:
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Write down the domain and range of ${f^{ - 1}}$ .
16N.2.SL.TZ0.S_1b:
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17M.1.AHL.TZ1.H_11a.ii:
Factorize ${x^2} + 3x + 2$ .
18M.2.AHL.TZ2.H_10a.i:
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18M.2.SL.TZ2.T_6b:
A teacher asks her students to make some observations about the curve.

Three students responded.
Nadia said “The x-intercept of the curve is between −1 and zero”.
Rick said “The curve is decreasing when x < 1 ”.
Paula said “The gradient of the curve is less than zero between x = 1 and x = 2 ”.

State the name of the student who made an incorrect observation.
19M.2.SL.TZ1.S_9d.i:
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18N.1.AHL.TZ0.H_3c.ii:
For the value of $a$ found in part (b), find an expression for ${g^{ - 1}}\left( x \right)$ .
18M.2.SL.TZ1.T_4b:
Using your value of k , find f ′(x).
18M.1.AHL.TZ2.H_10b.i:
Express $g\left( x \right)$ in the form $A + \frac{B}{{x - 2}}$ where A, B are constants.
18M.2.AHL.TZ2.H_10a.iii:
Explain why $f$ has no inverse on the given domain.
19M.2.SL.TZ1.S_9c:
Find the acute angle between $y = x$ and $L$ .
17M.1.AHL.TZ1.H_11e:
Sketch the graph of $y = f\left( {\left| x \right|} \right)$ .
17M.2.AHL.TZ1.H_12c:
Explain why $f$ is an even function.
17M.1.AHL.TZ2.H_2a:
Write down the range of $f$ .
16N.1.AHL.TZ0.H_3b:
find the value of $b$ .
18M.1.SL.TZ1.S_3c:
On the grid above, sketch the graph of f ⁻¹.
16N.2.AHL.TZ0.H_5b:
State the range of $f$ .
16N.1.SL.TZ0.T_9b:
Find the value of $b$ .
17M.2.SL.TZ2.S_6b:
On the following grid, sketch the graph of $(f \circ g)(x)$ , for $0 \leqslant x \leqslant 2.25$ .
17M.1.AHL.TZ2.H_2b:
Find an expression for ${f^{ - 1}}(x)$ .
17M.1.AHL.TZ1.H_11d:
Hence find the value of $p$ if $\int_0^1 {f(x){\text{d}}x = \ln (p)}$ .
17N.2.AHL.TZ0.H_10a.i:
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19M.1.AHL.TZ1.H_8a:
Write down the $x$ -coordinate of the point of inflexion on the graph of $y = f\left( x \right)$ .
17M.1.SL.TZ2.T_14c:
Calculate how much extra money Jashanti needs.
19M.2.SL.TZ1.T_4e:
This straight road crosses the highway and then carries on due north.

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17M.2.AHL.TZ1.H_12g.ii:
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18M.1.AHL.TZ2.H_10c:
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17N.1.AHL.TZ0.H_6a:
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18M.1.AHL.TZ1.H_9b.i:
Show that there is exactly one point of inflexion, B, on the graph of $y = f\left( x \right)$ .
18M.1.AHL.TZ2.H_2b:
Solve the equation $\frac{x}{2} + 1 = \left| {x - 2} \right|$ .
18M.1.AHL.TZ1.H_9a:
The graph of $y = f\left( x \right)$ has a local maximum at A. Find the coordinates of A.
17M.1.AHL.TZ1.H_11a.i:
Express ${x^2} + 3x + 2$ in the form ${(x + h)^2} + k$ .
16N.1.AHL.TZ0.H_3a:
state the value of $a$ and the value of $c$ ;
17N.2.AHL.TZ0.H_10c:
Find the coordinates of the point on the graph of $f$ where the normal to the graph is parallel to the line $y = - x$ .
17N.2.AHL.TZ0.H_10b:
Sketch the graph of $y = f(x)$ showing clearly the minimum point and any asymptotic behaviour.
20N.1.SL.TZ0.T_2a.i:
State, in the context of the question, what the value of $34.50$ represents.
20N.1.SL.TZ0.T_2a.ii:
State, in the context of the question, what the value of $8.50$ represents.
20N.1.SL.TZ0.T_2c:
Kaelani has $450 PGK$ .

Find the maximum number of large cheese pizzas that Kaelani can order from Olava’s Pizza Company.
17N.2.AHL.TZ0.H_10a.ii:
Determine the values of $x$ for which $f(x)$ is a decreasing function.
17M.1.AHL.TZ1.H_11c:
Show that $\frac{1}{{x + 1}} - \frac{1}{{x + 2}} = \frac{1}{{{x^2} + 3x + 2}}$ .
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_12d:
Explain why the inverse function ${f^{ - 1}}$ does not exist.
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:
Show that $(f \circ g)(x) = {x^4} - 4{x^2} + 3$ .
19M.2.AHL.TZ2.H_4b:
Hence solve ${\text{si}}{{\text{n}}^3}\,x + {\text{ln}}\,x - {\text{cos}}\,x - 1 < 0$ in the range 0 < $x$ ≤ 9.
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.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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