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