| Date | May 2009 | Marks available | 3 | Reference code | 09M.2.sl.TZ1.3 |
| Level | SL only | Paper | 2 | Time zone | TZ1 |
| Command term | Find | Question number | 3 | Adapted from | N/A |
Question
Let \(f(x) = \frac{{3x}}{2} + 1\) , \(g(x) = 4\cos \left( {\frac{x}{3}} \right) - 1\) . Let \(h(x) = (g \circ f)(x)\) .
Find an expression for \(h(x)\) .
Write down the period of \(h\) .
Write down the range of \(h\) .
Markscheme
attempt to form any composition (even if order is reversed) (M1)
correct composition \(h(x) = g\left( {\frac{{3x}}{2} + 1} \right)\) (A1)
\(h(x) = 4\cos \left( {\frac{{\frac{{3x}}{2} + 1}}{3}} \right) - 1\) \(\left( {4\cos \left( {\frac{1}{2}x + \frac{1}{3}} \right) - 1,4\cos \left( {\frac{{3x + 2}}{6}} \right) - 1} \right)\) A1 N3
[3 marks]
period is \(4\pi (12.6)\) A1 N1
[1 mark]
range is \( - 5 \le h(x) \le 3\) \(\left( {\left[ { - 5,3} \right]} \right)\) A1A1 N2
[2 marks]
Examiners report
The majority of candidates handled the composition of the two given functions well. However, a large number of candidates had difficulties simplifying the result correctly. The period and range of the resulting trig function was not handled well. If candidates knew the definition of "range", they often did not express it correctly. Many candidates correctly used their GDCs to find the period and range, but this approach was not the most efficient.
The majority of candidates handled the composition of the two given functions well. However, a large number of candidates had difficulties simplifying the result correctly. The period and range of the resulting trig function was not handled well. If candidates knew the definition of "range", they often did not express it correctly. Many candidates correctly used their GDCs to find the period and range, but this approach was not the most efficient.
The majority of candidates handled the composition of the two given functions well. However, a large number of candidates had difficulties simplifying the result correctly. The period and range of the resulting trig function was not handled well. If candidates knew the definition of "range", they often did not express it correctly. Many candidates correctly used their GDCs to find the period and range, but this approach was not the most efficient.
