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Date	May 2022	Marks available	2	Reference code	22M.1.AHL.TZ2.6
Level	Additional Higher Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 2
Command term	Show that	Question number	6	Adapted from	N/A

Question

A function $f$ is defined by $f (x) = x \sqrt{1 - x^{2}}$ where $- 1 \leq x \leq 1$ .

The graph of $y = f (x)$ is shown below.

Show that $f$ is an odd function.

[2]

a.

The range of $f$ is $a \leq y \leq b$ , where $a, b \in ℝ$ .

Find the value of $a$ and the value of $b$ .

[6]

b.

Markscheme

attempts to replace $x$ with $- x$ M1

$f (- x) = - x \sqrt{1 - {(- x)}^{2}}$

$= - x \sqrt{1 - {(- x)}^{2}} (= - f (x))$ A1

Note: Award M1A1 for an attempt to calculate both $f (- x)$ and $- f (- x)$ independently, showing that they are equal.
Note: Award M1A0 for a graphical approach including evidence that either the graph is invariant after rotation by $180 °$ about the origin or the graph is invariant after a reflection in the $y$ -axis and then in the $x$ -axis (or vice versa).

so $f$ is an odd function AG

[2 marks]

a.

attempts both product rule and chain rule differentiation to find $f' (x)$ M1

$f' (x) = x \times \frac{1}{2} \times (- 2 x) \times {(1 - x^{2})}^{- \frac{1}{2}} + {(1 - x^{2})}^{\frac{1}{2}} \times 1 (= \sqrt{1 - x^{2}} - \frac{x^{2}}{\sqrt{1 - x^{2}}})$ A1

$= \frac{1 - 2 x^{2}}{\sqrt{1 - x^{2}}}$

sets their $f' (x) = 0$ M1

$\Rightarrow x = \pm \frac{1}{\sqrt{2}}$ A1

attempts to find at least one of $f (\pm \frac{1}{\sqrt{2}})$ (M1)

Note: Award M1 for an attempt to evaluate $f (x)$ at least at one of their $f' (x) = 0$ roots.

$a = - \frac{1}{2}$ and $b = \frac{1}{2}$ A1

Note: Award A1 for $- \frac{1}{2} \leq y \leq \frac{1}{2}$ .

[6 marks]

b.

Examiners report

[N/A]

a.

[N/A]

b.

Syllabus sections

Topic 2—Functions » AHL 2.14—Odd and even functions, self-inverse, inverse and domain restriction

Show 22 related questions

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19M.1.AHL.TZ2.H_11b:
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19M.1.AHL.TZ2.H_11c:
Sketch the graphs of $y = f\left( x \right)$ and $y = g\left( x \right)$ on the same axes, clearly stating the points of intersection with any axes.
18N.2.AHL.TZ0.H_9c.i:
sketch the graph of $y = f\left( x \right)$ , showing clearly any axis intercepts and giving the equations of any asymptotes.
18N.1.AHL.TZ0.H_3a:
For $a = - \frac{\pi }{2}$ , sketch the graph of $y = g\left( x \right)$ . Indicate clearly the maximum and minimum values of the function.
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Hence or otherwise, find an expression for the derivative of ${f_n}(x)$ with respect to $x$ .
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Find the largest value of $a$ such that $f$ has an inverse function.
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18N.2.AHL.TZ0.H_9c.ii:
sketch the graph of $y = {f^{ - 1}}\left( x \right)$ , showing clearly any axis intercepts and giving the equations of any asymptotes.
19M.1.AHL.TZ2.H_11d:
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SPM.1.AHL.TZ0.9b:
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22M.1.AHL.TZ2.11b.i:
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22M.2.AHL.TZ1.6a:
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22M.2.AHL.TZ1.10b.i:
Show that the inverse function of $f$ is given by $f^{- 1} (x) = \sqrt{x^{2} + 1}$ .
19M.1.AHL.TZ2.H_11a:
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18N.2.AHL.TZ0.H_9d:
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21M.2.AHL.TZ2.12a:
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17N.1.AHL.TZ0.H_11a:
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19M.2.AHL.TZ1.H_4b:
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17N.1.AHL.TZ0.H_11d:
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18N.2.AHL.TZ0.H_9b:
Hence, or otherwise, find the coordinates of the point of inflexion on the graph of $y = f\left( x \right)$ .

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