9 The functions f and g are defined for all real values of \(x\) by
$$f ( x ) = ( 3 x - 2 ) ^ { 2 } + k \quad \text { and } \quad g ( x ) = 5 x - 1$$
where \(k\) is a constant.
- Given that the range of the function gf is \(\mathrm { gf } ( x ) \geqslant 39\), find the value of \(k\).
- For this value of \(k\), determine the range of the function fg .
- The function h is defined for all real values of \(x\) and is such that \(\mathrm { gh } ( x ) = 35 x + 19\). Find an expression for \(\mathrm { g } ^ { - 1 } ( x )\) and hence, or otherwise, find an expression for \(\mathrm { h } ( x )\).
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The diagram shows the circle with centre \(C ( - 4,5 )\) and radius \(\sqrt { 20 }\) units. The circle intersects the \(y\)-axis at the points \(A\) and \(B\). The size of angle \(A C B\) is \(\theta\) radians.