Show that \((f \circ g)(x) = {x^4} - 4{x^2} + 3\).

On the following grid, sketch the graph of \((f \circ g)(x)\), for \(0 \leqslant x \leqslant 2.25\).

The equation \((f \circ g)(x) = k\) has exactly two solutions, for \(0 \leqslant x \leqslant 2.25\). Find the possible values of \(k\).

attempt to form composite in either order     (M1)

eg\(\,\,\,\,\,\)\(f({x^2} - 2),{\text{ }}{({x^2} - 1)^2} - 2\)

\(({x^4} - 4{x^2} + 4) - 1\)     A1

\((f \circ g)(x) = {x^4} - 4{x^2} + 3\)     AG     N0

[2 marks]

   A1

A1A1     N3

Note:     Award A1 for approximately correct shape which changes from concave down to concave up. Only if this A1 is awarded, award the following:

A1 for left hand endpoint in circle and right hand endpoint in oval,

A1 for minimum in oval.

[3 marks]

evidence of identifying max/min as relevant points     (M1)

eg\(\,\,\,\,\,\)\(x = 0,{\text{ }}1.41421,{\text{ }}y =  - 1,{\text{ }}3\)

correct interval (inclusion/exclusion of endpoints must be correct)     A2     N3

eg\(\,\,\,\,\,\)\( - 1 < k \leqslant 3,{\text{ }}\left] { - 1,{\text{ 3}}} \right],{\text{ }}( - 1,{\text{ }}3]\)

[3 marks]