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Date May 2008 Marks available 8 Reference code 08M.1.hl.TZ1.8
Level HL only Paper 1 Time zone TZ1
Command term Find and State Question number 8 Adapted from N/A

Question

The functions f and g are defined as:

\[f(x) = {{\text{e}}^{{x^2}}},{\text{ }}x \geqslant 0\]

\[g(x) = \frac{1}{{x + 3}},{\text{ }}x \ne - 3.\]

(a)     Find \(h(x){\text{ where }}h(x) = g \circ f(x)\) .

(b)     State the domain of \({h^{ - 1}}(x)\) .

(c)     Find \({h^{ - 1}}(x)\) .

Markscheme

(a)     \(h(x) = g \circ f(x) = \frac{1}{{{{\text{e}}^{{x^2}}} + 3}},{\text{ }}(x \geqslant 0)\)     (M1)A1

 

(b)     \(0 < x \leqslant \frac{1}{4}\)     A1A1

Note: Award A1 for limits and A1 for correct inequality signs.

 

(c)     \(y = \frac{1}{{{{\text{e}}^{{x^2}}} + 3}}\)

\(y{{\text{e}}^{{x^2}}} + 3y = 1\)     M1

\({{\text{e}}^{{x^2}}} = \frac{{1 - 3y}}{y}\)     A1

\({x^2} = \ln \frac{{1 - 3y}}{y}\)     M1

\(x = \pm \sqrt {\ln \frac{{1 - 3y}}{y}} \)

\( \Rightarrow {h^{ - 1}}(x) = \sqrt {\ln \frac{{1 - 3x}}{x}} {\text{ }}\left( { = \sqrt {\ln \left( {\frac{1}{x} - 3} \right)} } \right)\)     A1

[8 marks]

Examiners report

Part (a) was correctly done by the vast majority of candidates. In contrast, only the very best students gave the correct answer to part (b). Part (c) was correctly started by a majority of candidates, but many did not realise that they needed to use logarithms and were careless about the use of notation

Syllabus sections

Topic 2 - Core: Functions and equations » 2.1 » Composite functions \(f \circ g\) .

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