June 2022 Paper 3 Q2

OCR MEICurrent spec6 marksFunctions (including |mod|)

2 The function \(\mathrm{f}(x) = \sqrt{x}\) is defined on the domain \(x \geqslant 0\).

The function \(\mathrm{g}(x) = 25 - x^2\) is defined on the domain \(\mathbb{R}\).

(a) Write down an expression for \(\mathrm{fg}(x)\). [1]

(b)

(i) Find the domain of \(\mathrm{fg}(x)\). [3]

(ii) Find the range of \(\mathrm{fg}(x)\). [2]

Scheme	Marks	AO
\(\sqrt{25 - x^2}\)	B1	1.1
	[1]

Mark scheme (b)(i)

Scheme	Marks	AO
\(25 - x^2 \geqslant 0\)	M1	1.1a
\(-5 \leqslant x \leqslant 5\) o.e.	B1	2.2a
	B1	2.5
	[3]

Notes

M1: Implied by B2

B1: At least one part of the domain correct, condone missing = for this mark, e.g. \(x < 5\)
Can score M0 B1

B1: Domain completely correct and correctly expressed
Eg allow \(x \geqslant -5\) and \(x \leqslant 5\) but not \(x \geqslant -5\) or \(x \leqslant 5\) or other ambiguous notation eg a comma
Condone \(x \leqslant \pm 5\) seen in working if final answer is correct

Mark scheme (b)(ii)

Scheme	Marks	AO
\(0 \leqslant \mathrm{fg}(x) \leqslant 5\) or \(0 \leqslant y \leqslant 5\)	B1	2.2a
	B1	2.5
	[2]

Notes

B1: At least one part of the range correct, condone missing = for this mark, e.g. \(y < 5\), allow use of \(\mathrm{f}(x)\) or \(\mathrm{g}(x)\) notation here but not \(x\)

B1: Range completely correct and correctly expressed
See notes in 2(b)(i)