Find $(g \circ f)(x)$ .

Write down ${g^{ - 1}}(x)$ .

Find $(f \circ {g^{ - 1}})(5)$ .

attempt to form composite     (M1)

e.g. $g(7 - 2x)$ , $7 - 2x + 3$

$(g \circ f)(x) = 10 - 2x$     A1     N2

[2 marks]

${g^{ - 1}}(x) = x - 3$     A1     N1

[1 mark]

METHOD 1

valid approach     (M1)

e.g. ${g^{ - 1}}(5)$ , $2$ , $f(5)$

$f(2) = 3$     A1     N2

METHOD 2

attempt to form composite of f and ${g^{ - 1}}$     (M1)

e.g. $(f \circ {g^{ - 1}})(x) = 7 - 2(x - 3)$ , $13 - 2x$

$(f \circ {g^{ - 1}})(5) = 3$     A1     N2

[2 marks]

A majority of candidates found success in the opening question. Common errors in (a) were to give $f \circ g$ or to multiply f by g. 

For (b) some gave the inverse as the reciprocal function $\frac{1}{{x + 3}}$ , or wrote $x = y + 3$ .

Most candidates chose to find a composite in (c), sometimes making simple errors when working with brackets and a negative sign. Only a handful used the more efficient $f(2) = 3$ . Additionally, it was not uncommon for candidates to give a correct substitution but not complete the result. Simple expressions such as $(7 - 2x) + 3$ should be finished as $10 - 2x$ .