June 2024 Paper 1 Q17

17 The function \(\mathrm{f}\) is defined by

\[\mathrm{f}(x) = |x| + 1 \text{ for } x \in \mathbb{R}\]

The function \(\mathrm{g}\) is defined by

\[\mathrm{g}(x) = \ln x\]

where \(\mathrm{g}\) has its greatest possible domain.

(a) Using set notation, state the range of \(\mathrm{f}\) [2 marks]

(b) State the domain of \(\mathrm{g}\) [1 mark]

(i) Write down an expression for \(\mathrm{h}(x)\) in terms of \(x\) [1 mark]

(ii) Determine if \(\mathrm{h}\) has an inverse.

Fully justify your answer. [2 marks]

Mark scheme (a)

Scheme	Marks	AO
Deduces correct region eg \(\mathrm{f}(x) \geqslant 1\) or \(y \geqslant 1\) Condone \(\mathrm{f}(x) \gt 1\) or \(y \gt 1\)	M1	2.2a
Obtains correct answer in set notation eg \(\{x : x \geqslant 1\}\) \(\{\mathrm{f}(x) : \mathrm{f}(x) \geqslant 1\}\) \([1, \infty)\)	A1	2.5
	(2)

\[\{y : y \geqslant 1\}\]

Scheme	Marks	AO
Obtains \(\{x : x \gt 0\}\) or \((0, \infty)\) Accept \(\{y : y \gt 0\}\) but not \(y \gt 0\)	B1	1.1b
	(1)

Scheme	Marks	AO
(i) Obtains \(\ln\left(\|x\| + 1\right)\) Accept \(\ln\left\|\|x\| + 1\right\|\) ISW	B1	1.1b
	(1)
(ii) States that \(\mathrm{h}\) does not have an inverse And States that \(\mathrm{h}\) is not one-to-one or that \(\mathrm{h}\) is many-to-one	E1	2.2a
Explains why \(\mathrm{h}\) is not one-to-one or why \(\mathrm{h}\) is many-to-one For example Gives two \(x\) values such that \(\mathrm{h}(x_1) = \mathrm{h}(x_2)\) Or Explains that using the positive and negative of the same \(x\)-value will result in the same \(y\)-value Or Sketches a graph of y=h(x) with a horizontal line meeting the curve in two places. eg Or States that \(\mathrm{f}\) is many-to-one and that \(\mathrm{h}\) is a function of \(\mathrm{f}\).	E1	2.4
	(2)
	(6 marks)