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Date	May 2013	Marks available	3	Reference code	13M.1.sl.TZ2.1
Level	SL only	Paper	1	Time zone	TZ2
Command term	Find	Question number	1	Adapted from	N/A

Question

Let \(f(x) = 4x - 2\) and \(g(x) = - 2{x^2} + 8\) .

Find \({f^{ - 1}}(x)\) .

[3]

Find \((f \circ g)(1)\) .

[3]

Markscheme

interchanging \(x\) and \(y\) (seen anywhere) (M1)

eg \(x = 4y - 2\)

evidence of correct manipulation (A1)

eg \(x + 2 = 4y\)

\({f^{ - 1}}(x) = \frac{{x + 2}}{4}\) (accept \(y = \frac{{x + 2}}{4}\) , \(\frac{{x + 2}}{4}\) , \({f^{ - 1}}(x) = \frac{1}{4}x + \frac{1}{2}\) A1 N2

[3 marks]

METHOD 1

attempt to substitute \(1\) into \(g(x)\) (M1)

eg \(g(1) = - 2 \times {1^2} + 8\)

\(g(1) = 6\) (A1)

\(f(6) = 22\) A1 N3

METHOD 2

attempt to form composite function (in any order) (M1)

eg \((f \circ g)(x) = 4( - 2{x^2} + 8) - 2\) \(( = - 8{x^2} + 30)\)

correct substitution

eg \((f \circ g)(1) = 4( - 2 \times {1^2} + 8) - 2\) , \( - 8 + 30\)

\(f(6) = 22\) A1 N3

[3 marks]

Examiners report

The overwhelming majority of candidates answered both parts of this question correctly. There were a few who seemed unfamiliar with the inverse notation and answered part (a) with the derivative or the reciprocal of the function.

The overwhelming majority of candidates answered both parts of this question correctly. A few candidates made arithmetic errors in part (b) which kept them from finding the correct answer.

Syllabus sections

Topic 2 - Functions and equations » 2.1 » Identity function. Inverse function \({f^{ - 1}}\) .

Show 45 related questions

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08M.1.sl.TZ1.7a: Find \({f^{ - 1}}(x)\) .
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14M.1.sl.TZ2.3a(ii): Write down the value of \({f^{ - 1}}(1)\).
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16M.1.sl.TZ1.1b: Find \((f \circ g)(x)\).
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12M.1.sl.TZ1.9a: Find p .
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12M.1.sl.TZ2.2a: Find \({f^{ - 1}}(x)\) .
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11M.2.sl.TZ2.1b: Find \({h^{ - 1}}(x)\) .
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09N.1.sl.TZ0.7a: Given that \({f^{ - 1}}(1) = 8\) , find the value of \(k\) .
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13M.1.sl.TZ1.5a: Find \({f^{ - 1}}(2)\) .
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