Sketch the graph of $y = {f^{ - 1}}(x)$ on the same axes.

State the range of ${f^{ - 1}}$.

Given that $f(x) = \ln (ax + b),{\text{ }}x > 1$, find the value of $a$ and the value of $b$.

(a)     

shape with y-axis intercept (0, 4)     A1

Note:     Accept curve with an asymptote at $x = 1$ suggested.

correct asymptote $y = 1$     A1

[2 marks]

range is ${f^{ - 1}}(x) > 1{\text{ (or }}\left] {1,{\text{ }}\infty } \right[)$     A1

Note:     Also accept $\left] {1,{\text{ 10}}} \right]$ or $\left] {1,{\text{ 10}}} \right[$.

Note:     Do not allow follow through from incorrect asymptote in (a).

[1 mark]

$(4,{\text{ }}0) \Rightarrow \ln (4a + b) = 0$     M1

$\Rightarrow 4a + b = 1$     A1

asymptote at $x = 1 \Rightarrow a + b = 0$     M1

$\Rightarrow a = \frac{1}{3},{\text{ }}b =  - \frac{1}{3}$     A1

[4 marks]

A number of candidates were able to answer a) and b) correctly but found part c) more challenging. Correct sketches for the inverse were seen, but with a few missing a horizontal asymptote. The range in part b) was usually seen correctly. In part c), only a small number of very good candidates were able to gain full marks. A large number used the point $(4,{\text{ 0)}}$ to form the equation $4a + b = 1$ but were unable (or did not recognise the need) to use the asymptote to form a second equation.

A number of candidates were able to answer a) and b) correctly but found part c) more challenging. Correct sketches for the inverse were seen, but with a few missing a horizontal asymptote. The range in part b) was usually seen correctly. In part c), only a small number of very good candidates were able to gain full marks. A large number used the point $(4,{\text{ 0)}}$ to form the equation $4a + b = 1$ but were unable (or did not recognise the need) to use the asymptote to form a second equation.

A number of candidates were able to answer a) and b) correctly but found part c) more challenging. Correct sketches for the inverse were seen, but with a few missing a horizontal asymptote. The range in part b) was usually seen correctly. In part c), only a small number of very good candidates were able to gain full marks. A large number used the point $(4,{\text{ 0)}}$ to form the equation $4a + b = 1$ but were unable (or did not recognise the need) to use the asymptote to form a second equation.