(a) The functions \(f(x)\) and \(g(x)\) are defined by\[\begin{aligned} f(x) &= \sin x^\circ, \\ g(x) &= x - 90. \end{aligned}\]Find the value of \(fg(120)\). [2]
(b) The diagram shows a sketch of the graph of \(y = h(x)\).On the same diagram, sketch the graph of \(y = -h(x)\). Write down the coordinates of the maximum and minimum points of \(y = -h(x)\). [2]
(c) The function \(k(x)\) is defined by\[k(x) = x^3 - 23.\]Solve \(k^{-1}(x) = 5\). [4]
Mark scheme (a)
Scheme
Marks
Comments
\(\sin 30\)
M1
or \(fg(x) = \sin(x - 90)\)
½ or 0.5
A1
Mark scheme (b)
Scheme
Marks
Comments
Reflection in the \(x\)-axis
B1
Mark intent
\((-2, 1)\) and \((-6, -0.5)\) indicated
B1
CAO
Mark scheme (c)
Scheme
Marks
Comments
\(\left[k^{-1}(x) =\right]\ \sqrt[3]{x + 23}\)
B2
Award B1 for \(x = \sqrt[3]{y + 23}\) oe, unless \(x\) and \(y\) interchanged later or SC1 for \(\left(y \text{ or } k^{-1}(x) =\right) \sqrt[3]{x - 23}\) oe
\(x + 23 = 5^3\) or better
M1
FT ‘their inverse function’ if possible; must be of equivalent difficulty; M0 for e.g. \(\dfrac{1}{x^3 - 23} = 5\)
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