18N.1.AHL.TZ0.H_3

pestleMathematicsAIHLPaper 118N· sl-2-2-functions-notation-domain-range-and-inverse-as-reflectionsource ↗

Consider the function $g\left( x \right) = 4\,{\text{cos}}\,x + 1$ , $a \leqslant x \leqslant \frac{\pi }{2}$ where $a < \frac{\pi }{2}$ .

For $a = - \frac{\pi }{2}$ , sketch the graph of $y = g\left( x \right)$ . Indicate clearly the maximum and minimum values of the function.

[3]

Write down the least value of $a$ such that $g$ has an inverse.

[1]

For the value of $a$ found in part (b), write down the domain of ${g^{ - 1}}$ .

[1]

c.i.

For the value of $a$ found in part (b), find an expression for ${g^{ - 1}}\left( x \right)$ .

[2]

c.ii.

Markscheme / solution

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

concave down and symmetrical over correct domain A1

indication of maximum and minimum values of the function (correct range) A1A1

[3 marks]

$a$ = 0 A1

Note: Award A1 for $a$ = 0 only if consistent with their graph.

[1 mark]

$1 \leqslant x \leqslant 5$ A1

Note: Allow FT from their graph.

[1 mark]

c.i.

$y = 4\,{\text{cos}}\,x + 1$

$x = 4\,{\text{cos}}\,y + 1$

$\frac{{x - 1}}{4} = {\text{cos}}\,y$ (M1)

$\Rightarrow y = {\text{arccos}}\left( {\frac{{x - 1}}{4}} \right)$

$\Rightarrow {g^{ - 1}}\left( x \right) = {\text{arccos}}\left( {\frac{{x - 1}}{4}} \right)$ A1

[2 marks]

c.ii.

Examiners’ report

[N/A]

c.i.

[N/A]

c.ii.