- The functions f and g are defined by
$$\begin{array} { l l }
f ( x ) = 4 - 3 x ^ { 2 } & x \in \mathbb { R }
g ( x ) = \frac { 5 } { 2 x - 9 } & x \in \mathbb { R } , x \neq \frac { 9 } { 2 }
\end{array}$$
- Find fg(2)
- Find \(\mathrm { g } ^ { - 1 }\)
- Find \(\mathrm { gf } ( x )\), giving your answer as a simplified fraction.
- Deduce the range of \(\operatorname { gf } ( x )\).
The function h is defined by
$$h ( x ) = 2 x ^ { 2 } - 6 x + k \quad x \in \mathbb { R }$$
where \(k\) is a constant.
- Find the range of values of \(k\) for which the equation
$$\mathrm { f } ( x ) = \mathrm { h } ( x )$$
has no real solutions.