June 2023 Paper 2 Q12

OCR MEICurrent spec4 marksFunctions (including |mod|)

12 It is given that

\(\mathrm{f}(x) = \pm\frac{1}{\sqrt{x}},\ x > 0\)
\(\mathrm{g}(x) = \frac{x}{x-3},\ x > 3\)
\(\mathrm{h}(x) = x^2 + 2,\ x \in \mathbb{R}\).

(a) Explain why \(\mathrm{f}(x)\) is not a function. [1]

(b) Find \(\mathrm{gh}(x)\). [2]

Mark scheme (a)

Scheme	Marks	AO
because it’s neither a one-to-one nor a many-to-one (mapping) do not allow eg because it’s neither a one-to-one nor a many-to-one function	B1	2.4
	[1]

Notes

B1: allow because it’s one-to-many (mapping)
allow eg because each value of \(x\) is mapped to two values oe
do not allow
eg because it’s a one-to-many function

Scheme	Marks	AO
\(\dfrac{x^2+2}{x^2+2-3}\)	M1	1.1
\(\dfrac{x^2+2}{x^2-1}\) or \(\dfrac{x^2+2}{(x-1)(x+1)}\)	A1	1.1
	[2]

Scheme	Marks	AO
\(\|x\| > 1\) or \(x < -1\) or \(x > 1\) or \(x < -1,\ x > 1\) or \(x < -1 \cup x > 1\)	B1	1.1
	[1]