2015 SAJC P1 Q9

The function \(\mathrm{g}\), with domain \(\{x\in \mathbb{R}:1\le x\le 6\}\), has six of its function values given in the table below.

1. Use the table to explain why \(\mathrm{g}\) does not have an inverse function.[2]
2. Find \({{\mathrm{g}}^{3}}\left( 3 \right)\). Hence find the set of all positive integers \(n\) for which \({{\mathrm{g}}^{n}}\left( 3 \right)=4\).[3]

It is also known that for \(k=1\), \(2\), \(3\), \(4\), \(5\)

\(\mathrm{g}\left( x \right)=\mathrm{g}\left( k \right)+\left( \mathrm{g}\left( k+1 \right)-\mathrm{g}\left( k \right) \right)\left( x-k \right)\) for \(k < x < k+1\)

3. Evaluate \(\mathrm{g}\left( 1.5 \right)\)and \(\mathrm{g}\left( 2.7 \right)\).[2]
4. Sketch the graph of \(y=\mathrm{g}\left( x \right)\).[2]
5. Find the range of values of \(k\) for which the equation \(\mathrm{g}\left( x \right)=k\) has four real distinct roots.[1]