Find ${f^{ - 1}}(x)$.

Let $g$ be a function so that $(f \circ g)(x) = 8{x^6}$. Find $g(x)$.

interchanging $x$ and $y$ (seen anywhere)     (M1)

eg$\;\;\;x = {(y - 5)^3}$

evidence of correct manipulation     (A1)

eg$\;\;\;y - 5 = \sqrt[3]{x}$

${f^{ - 1}}(x) = \sqrt[3]{x} + 5\;\;\;({\text{accept }}5 + {x^{\frac{1}{3}}},{\text{ }}y = 5 + \sqrt[3]{x})$     A1     N2

Notes:     If working shown, and they do not interchange $x$ and $y$, award A1A1M0 for $\sqrt[3]{y} + 5$.

If no working shown, award N1 for $\sqrt[3]{y} + 5$.

METHOD 1

attempt to form composite (in any order)     (M1)

eg$\;\;\;g\left( {{{(x - 5)}^3}} \right),{\text{ }}{\left( {g(x) - 5} \right)^3} = 8{x^6},{\text{ }}f(2{x^2} + 5)$

correct working     (A1)

eg$\;\;\;g - 5 = 2{x^2},{\text{ }}{\left( {(2{x^2} + 5) - 5} \right)^3}$

$g(x) = 2{x^2} + 5$     A1     N2

METHOD 2

recognising inverse relationship     (M1)

eg$\;\;\;{f^{ - 1}}(8{x^6}) = g(x),{\text{ }}{f^{ - 1}}(f \circ g)(x) = {f^{ - 1}}(8{x^6})$

correct working

eg$\;\;\;g(x) = \sqrt[3]{{(8{x^6})}} + 5$     (A1)

$g(x) = 2{x^2} + 5$     A1     N2