find the range of \(f\).

find the value of \(x\) given that \(f (x) =162\).

\(f (-2) = 2 \times 3^{-2}\)     (M1)

\(= \frac{{2}}{{9}}(0.222)\)     (A1)

\(f (5) = 2 \times 3^5\)

\(= 486\)     (A1)

\({\text{Range }}\frac{2}{9} \leqslant f(x) \leqslant 486\)   OR   \(\left[ {\frac{2}{9},{\text{ }}486} \right]\)     (A1)     (C4)

Note: Award (M1) for correct substitution of –2 or 5 into \(f (x)\), (A1)(A1) for each correct end point.

[4 marks]

\(2 \times 3^x = 162\)     (M1)

\(x = 4\)     (A1)     (C2)

[2 marks]

Part (a) proved to be difficult to gain the maximum marks as, although candidates could find the end points, they did not seem to be able to identify the range of the function. Many students gave a list of values for the range, which indicates that this concept was not understood well.

This question was generally answered well in part (b).