June 2023 Paper 2 Q15
15 The function g is given by \(\quad \mathrm{g}(x) = a \times b^x \quad\) where \(a\) and \(b\) are constants.
The domain of the function is \(\quad -1 \leqslant x \leqslant 2\)
\(P\left(0, \dfrac{1}{2}\right)\) and \(Q\left(1, \dfrac{3}{2}\right)\) are points on the graph \(\quad y = \mathrm{g}(x)\)
Not drawn accurately
Work out the range of the function. [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(\dfrac{1}{2} = a \times b^0\) or \(a = \dfrac{1}{2}\) or \(\dfrac{3}{2} = a \times b^{(1)}\) | M1 | oe eg \(a \times 1 = 0.5\) or \(ab = 1.5\) |
| (\(a =\)) \(\dfrac{1}{2}\) and (\(b =\)) 3 | A1 | oe values may be implied eg \(y = \dfrac{1}{2} \times 3^x\) |
| their \(\dfrac{1}{2} \times (\text{their } 3)^{-1}\) or \(\dfrac{1}{2} \times \dfrac{1}{3}\) or \(\dfrac{1}{6}\) and their \(\dfrac{1}{2} \times (\text{their } 3)^2\) or \(\dfrac{1}{2} \times 3^2\) or \(\dfrac{9}{2}\) | M1 | oe neither their \(\dfrac{1}{2}\) nor their 3 can be 0 or 1 \(\dfrac{1}{6}\) and \(\dfrac{9}{2}\) is M1A1M1 ignore other calculations or values |
| \(\dfrac{1}{6} \leqslant \mathrm{g}(x) \leqslant \dfrac{9}{2}\) or both of \(\mathrm{g}(x) \geqslant \dfrac{1}{6} \qquad \mathrm{g}(x) \leqslant \dfrac{9}{2}\) | A1ft | oe eg \(\left[\dfrac{1}{6}, \dfrac{9}{2}\right]\) only ft M1A0M1 |
Additional guidance
| Ignore words if answer given as separate inequalities eg1 \(\mathrm{g}(x) \geqslant \dfrac{1}{6}\) or \(\mathrm{g}(x) \leqslant \dfrac{9}{2}\) eg2 \(\mathrm{g}(x) \geqslant \dfrac{1}{6}\) and \(\mathrm{g}(x) \leqslant 4.5\) | M1A1M1A1 M1A1M1A1 |
| \(a = \dfrac{1}{2}\) and \(b = 2\) \(\dfrac{1}{4} \leqslant \mathrm{g}(x) \leqslant 2\) | M1A0 M1A1ft |
| Condone \(\mathrm{g}(x)\) replaced by eg \(y\) or g or g\(x\) or \(\mathrm{f}(x)\) or f or f\(x\) or \(a \times b^x\) | |
| Do not allow \(\mathrm{g}(x)\) to be replaced by \(x\) for the final mark eg \(\dfrac{1}{6} \leqslant x \leqslant \dfrac{9}{2}\) | M1A1M1A0 |
| Equivalent inequalities may be seen eg \(\dfrac{9}{2} \geqslant \mathrm{g}(x) \geqslant \dfrac{1}{6}\) | M1A1M1A1 |
| Inequality symbols must be correct eg \(\dfrac{1}{6} \lt \mathrm{g}(x) \leqslant \dfrac{9}{2}\) or \(\dfrac{1}{6} \leqslant \mathrm{g}(x) \geqslant \dfrac{9}{2}\) | M1A1M1A0 |
| \(\dfrac{1}{6} \leqslant \mathrm{g}(x) \leqslant \dfrac{9}{2}\) in working with list of integers on answer line | M1A1M1A0 |
| Only a list of integers | M0A0M0A0 |
| Allow 0.16(…) or 0.17 for \(\dfrac{1}{6}\) | |
| Correct answer followed by incorrect further work eg \(\dfrac{1}{6} \leqslant \mathrm{g}(x) \leqslant \dfrac{9}{2}\) and \(\left(\dfrac{9}{2} - \dfrac{1}{6} =\right) \dfrac{13}{3}\) | M1A1M1A0 |