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Question
Let \(f(x) = {\log _3}\frac{x}{2} + {\log _3}16 – {\log _3}4\) , for \(x > 0\) .
Show that \(f(x) = {\log _3}2x\) .
Find the value of \(f(0.5)\) and of \(f(4.5)\) .
The function f can also be written in the form \(f(x) = \frac{{\ln ax}}{{\ln b}}\) .
(i) Write down the value of a and of b .
(ii) Hence on graph paper, sketch the graph of f , for \( – 5 \le x \le 5\) , \( – 5 \le y \le 5\) , using a scale of 1 cm to 1 unit on each axis.
(iii) Write down the equation of the asymptote.
Write down the value of \({f^{ – 1}}(0)\) .
The point A lies on the graph of f . At A, \(x = 4.5\) .
On your diagram, sketch the graph of \({f^{ – 1}}\) , noting clearly the image of point A.
Answer/Explanation
Markscheme
combining 2 terms (A1)
e.g. \({\log _3}8x – {\log _3}4\) , \({\log _3}\frac{1}{2}x + {\log _3}4\)
expression which clearly leads to answer given A1
e.g. \({\log _3}\frac{{8x}}{4}\) , \({\log _3}\frac{{4x}}{2}\)
\(f(x) = {\log _3}2x\) AG N0
[2 marks]
attempt to substitute either value into f (M1)
e.g. \({\log _3}1\) , \({\log _3}9\)
\(f(0.5) = 0\) , \(f(4.5) = 2\) A1A1 N3
[3 marks]
(i) \(a = 2\) , \(b = 3\) A1A1 N1N1
(ii)
A1A1A1 N3
Note: Award A1 for sketch approximately through \((0.5 \pm 0.1{\text{, }}0 \pm 0.1)\) , A1 for approximately correct shape, A1 for sketch asymptotic to the y-axis.
(iii) \(x = 0\) (must be an equation) A1 N1
[6 marks]
\({f^{ – 1}}(0) = 0.5\) A1 N1
[1 mark]
A1A1A1A1 N4
Note: Award A1 for sketch approximately through \((0 \pm 0.1{\text{, }}0.5 \pm 0.1)\) , A1 for approximately correct shape of the graph reflected over \(y = x\) , A1 for sketch asymptotic to x-axis, A1 for point \((2 \pm 0.1{\text{, }}4.5 \pm 0.1)\) clearly marked and on curve.
[4 marks]