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Date	May 2022	Marks available	2	Reference code	22M.2.AHL.TZ1.6
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Show that	Question number	6	Adapted from	N/A

Question

Consider the function $f (x) = 2^{x} - \frac{1}{2^{x}}, x \in ℝ$ .

The function $g$ is given by $g (x) = \frac{x - 1}{x^{2} - 2 x - 3}$ , where $x \in ℝ, x \neq - 1, x \neq 3$ .

Show that $f$ is an odd function.

[2]

a.

Solve the inequality $f (x) \geq g (x)$ .

[4]

b.

Markscheme

attempt to replace $x$ with $- x$ M1

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

EITHER

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

OR

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

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.

attempt to find at least one intersection point (M1)

$x = - 1.26686 \dots, x = 0.177935 \dots, x = 3.06167 \dots$

$x = - 1.27, x = 0.178, x = 3.06$

$- 1.27 \leq x \leq - 1,$ A1

$0.178 \leq x < 3,$ A1

$x \geq 3.06$ A1

[4 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

22M.1.AHL.TZ2.6a:
Show that $f$ is an odd function.
22M.1.AHL.TZ2.11b.ii:
State the domain of $g^{- 1}$ .
19M.1.AHL.TZ2.H_11b:
State the range of $g$ .
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.
19M.2.AHL.TZ1.H_4a:
Write down the range of $f$ .
17N.1.AHL.TZ0.H_11c:
Hence or otherwise, find an expression for the derivative of ${f_n}(x)$ with respect to $x$ .
SPM.1.AHL.TZ0.9a:
Find the largest value of $a$ such that $f$ has an inverse function.
17N.1.AHL.TZ0.H_11b:
By using mathematical induction, prove that

${f_n}(x) = \frac{{\sin {2^{n + 1}}x}}{{{2^n}\sin 2x}},{\text{ }}x \ne \frac{{m\pi }}{2}$ where $m \in \mathbb{Z}$ .
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:
Find the coordinates of P.
SPM.1.AHL.TZ0.9b:
For this value of $a$ , find an expression for ${f^{ - 1}}\left( x \right)$ , stating its domain.
22M.1.AHL.TZ2.11b.i:
Show that $g^{- 1} (x) = 1 + \frac{\sqrt{4 x^{2} + x}}{x}$ .
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:
Describe the transformation by which $f\left( x \right)$ is transformed to $g\left( x \right)$ .
18N.2.AHL.TZ0.H_9d:
Hence, or otherwise, solve the inequality $f\left( x \right) > {f^{ - 1}}\left( x \right)$ .
21M.2.AHL.TZ2.12a:
Show that $f$ is an even function.
17N.1.AHL.TZ0.H_11a:
Determine whether ${f_n}$ is an odd or even function, justifying your answer.
19M.2.AHL.TZ1.H_4b:
Find $f'\left( x \right)$ , stating its domain.
17N.1.AHL.TZ0.H_11d:
Show that, for $n > 1$ , the equation of the tangent to the curve $y = {f_n}(x)$ at $x = \frac{\pi }{4}$ is $4x - 2y - \pi = 0$ .
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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Topic 2—Functions