For the function \(f(x)\), it is known that  \(f(3)=1, f^{\prime}(3)=2\)  and  \(f^{\prime \prime}(3)=4\).

Let  \(g(x)=f^{-1}(x)\).

What is the value of \(g^{\prime \prime}(1)\) ?

1. \(\dfrac{1}{4}\)
2. \(-\dfrac{1}{4}\)
3. \(-\dfrac{1}{2}\)
4. \(-1\)

Let  \(f(x)=2 x+\ln x\), for \(x>0\).

1. Explain why the inverse of \(f(x)\) is a function.  (1 mark)
   

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2. Let  \(g(x)=f^{-1}(x)\). By considering the value of \(f(1)\), or otherwise, evaluate \(g^{\prime}(2)\).  (2 mark)
   

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A given function  `f(x)`  has an inverse  `f^{-1}(x)`.

The derivatives of  `f(x)`  and  `f^{-1}(x)`  exist for all real numbers `x`.

The graphs  `y=f(x)`  and  `y=f^{-1}(x)`  have at least one point of intersection.

Which statement is true for all points of intersection of these graphs?

1. All points of intersection lie on the line  `y=x`.
2. None of the points of intersection lie on the line  `y=x`.
3. At no point of intersection are the tangents to the graphs parallel.
4. At no point of intersection are the tangents to the graphs perpendicular.

The polynomial  `g(x) = x^3 + 4x - 2`  passes through the point (1, 3).

Find the gradient of the tangent to  `f(x) = xg^(-1)(x)`  at the point where  `x = 3`.  (2 marks)