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Date	May 2022	Marks available	4	Reference code	22M.2.AHL.TZ1.6
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Solve	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 » SL 2.10—Solving equations graphically and analytically

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