7 The function f is defined for \(x > 0\) by \(\mathrm { f } ( x ) = \ln x\) and the function g is defined for all real values of \(x\) by \(\mathrm { g } ( x ) = x ^ { 2 } + 8\).
- Find the exact, positive value of \(x\) which satisfies the equation \(\operatorname { fg } ( x ) = 8\).
- State which one of f and g has an inverse and define that inverse function.
- Find the exact value of the gradient of the curve \(y = \operatorname { gf } ( x )\) at the point with \(x\)-coordinate \(\mathrm { e } ^ { 3 }\).
- Use Simpson's rule with four strips to find an approximate value of
$$\int _ { - 4 } ^ { 4 } \mathrm { fg } ( x ) \mathrm { d } x$$
giving your answer correct to 3 significant figures.