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]