SPM.1.SL.TZ0.5

pestleMathematicsAASLPaper 1SPM· sl-2-5-composite-functions-identity-finding-inversesource ↗

The functions $f$ and $g$ are defined such that $f\left( x \right) = \frac{{x + 3}}{4}$ and $g\left( x \right) = 8x + 5$ .

Show that $\left( {g \circ f} \right)\left( x \right) = 2x + 11$ .

[2]

Given that ${\left( {g \circ f} \right)^{ - 1}}\left( a \right) = 4$ , find the value of $a$ .

[3]

Markscheme / solution

attempt to form composition M1

correct substitution $g\left( {\frac{{x + 3}}{4}} \right) = 8\left( {\frac{{x + 3}}{4}} \right) + 5$ A1

$\left( {g \circ f} \right)\left( x \right) = 2x + 11$ AG

[2 marks]

attempt to substitute 4 (seen anywhere) (M1)

correct equation $a = 2 \times 4 + 11$ (A1)

$a$ = 19 A1

[3 marks]

Examiners’ report

[N/A]