6. The function f is defined by
$$\mathrm { f } : x \rightarrow \mathrm { e } ^ { 2 x } + k ^ { 2 } , \quad x \in \mathbb { R } , \quad k \text { is a positive constant. }$$
- State the range of f .
- Find \(\mathrm { f } ^ { - 1 }\) and state its domain.
The function g is defined by
$$g : x \rightarrow \ln ( 2 x ) , \quad x > 0$$
- Solve the equation
$$\mathrm { g } ( x ) + \mathrm { g } \left( x ^ { 2 } \right) + \mathrm { g } \left( x ^ { 3 } \right) = 6$$
giving your answer in its simplest form.
- Find \(\mathrm { fg } ( x )\), giving your answer in its simplest form.
- Find, in terms of the constant \(k\), the solution of the equation
$$\mathrm { fg } ( x ) = 2 k ^ { 2 }$$