O LevelAdditional MathsCambridgeExam QuestionsAlgebra & FunctionsFunctions

Functions (Cambridge O Level Additional Maths)

Exam Questions

1 hour13 questions
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1a
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q8-0606-m20-qp-22-additional-maths

The diagram shows the graph of straight f open parentheses x close parentheses equals a space cos space b x space plus c space for space 0 less or equal than x less or equal than fraction numerator 8 pi over denominator 3 end fractionradians.

Explain why straight f is a function.

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1b
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Write down the range of straight f.

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2
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straight g open parentheses x close parentheses equals 3 plus 1 over x space for space x greater or equal than 1.

(i)
Find an expression for straight g to the power of negative 1 end exponent space left parenthesis x right parenthesis.
[2]

(ii)
Write down the range of straight g to the power of negative 1 end exponent.
[1]

(iii)
Find the domain of straight g to the power of negative 1 end exponent.
[2]

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3a
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It is given that straight f open parentheses x close parentheses equals 5 space ln open parentheses 2 x plus 3 close parentheses space for space x greater than negative 3 over 2.

Write down the range of straight f.

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3b
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Findspace straight f to the power of negative 1 end exponent and state its domain.

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3c
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On the axes below, sketch the graph of y space equals space straight f left parenthesis x right parenthesis and the graph of y space equals space straight f to the power of negative 1 end exponent space left parenthesis x right parenthesis. Label each curve and state the intercepts on the coordinate axes.

q7c-0606-w20-qp-11-additional-maths

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4
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straight f left parenthesis x right parenthesis space equals space 4 space ln left parenthesis 2 x minus 1 right parenthesis

(i)
Write down the largest possible domain for the function straight f.

[1]

(ii)
Find straight f to the power of negative 1 end exponent space left parenthesis x right parenthesis and its domain.

[3]

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5
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straight h open parentheses x close parentheses space equals 2 space ln open parentheses 3 x minus 1 close parentheses space for space x greater or equal than 2 over 3.
The graph of y space equals space straight h left parenthesis x right parenthesis intersects the line y space equals space x at two distinct points. On the axes below, sketch the graph of y space equals space straight h left parenthesis x right parenthesis and hence sketch the graph of y space equals space straight h to the power of negative 1 end exponent space left parenthesis x right parenthesis.

q10b-0606-m20-qp-22-additional-maths

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1
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It is given that space straight h left parenthesis x right parenthesis space equals space a space plus space b over x squared, where a and b are non-zero constants.

(i)
Explain why – 2 space less-than or slanted equal to space x space less-than or slanted equal to space 2 is not a suitable domain for straight h left parenthesis x right parenthesis.

(ii)
Given that straight h left parenthesis 1 right parenthesis space equals space 4 and straight h apostrophe left parenthesis 1 right parenthesis space equals space 16 , find the values of a and b.

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2a
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straight f space colon space x space rightwards arrow from bar space left parenthesis 2 x plus 3 right parenthesis squared space for space x space greater than space 0


Find the range of straight f.

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2b
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Explain why straight f has an inverse.

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2c
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Find space straight f to the power of negative 1 end exponent.

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2d
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State the domain of straight f to the power of negative 1 end exponent.

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2e
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Given that straight g space colon space x space rightwards arrow from bar space ln left parenthesis x plus 4 right parenthesis space for space x space greater than space 0, find the exact solution of fg left parenthesis x right parenthesis space equals space 49.

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3
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straight g left parenthesis x right parenthesis space equals space x plus 5 space for space x space space element of straight real numbers
straight h open parentheses x close parentheses equals space square root of 2 x minus 3 end root space space for space x greater or equal than 3 over 2

Solve gh left parenthesis x right parenthesis space equals space 7.

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4a
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The functions straight f and straight g are defined as follows.

straight f left parenthesis x right parenthesis space equals space x squared space plus 4 x space space for space space x space element of space R space space straight g left parenthesis x right parenthesis space equals space 1 plus straight e to the power of 2 x end exponent space space for space space space x space element of space R

Find the range of straight f.

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4b
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Write down the range of straight g.

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4c
Sme Calculator4 marks

Find the exact solution of the equation fg left parenthesis x right parenthesis space equals space 21, giving your answer as a single logarithm

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5a
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straight f left parenthesis x right parenthesis space equals space x squared space plus 2 x minus 3 space for space x space greater or equal than negative 1

Given that the minimum value of x squared space plus 2 x minus 3 occurs when x space equals negative 1, explain why straight f left parenthesis x right parenthesis has an inverse.

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5b
Sme Calculator4 marks

On the axes below, sketch the graph of y space equals space straight f left parenthesis x right parenthesis and the graph of y space equals space straight f to the power of negative 1 end exponent space left parenthesis x right parenthesis. Label each graph and state the intercepts on the coordinate axes.

q6b-0606-w20-qp-12-additional-maths

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1
Sme Calculator9 marks

straight f left parenthesis x right parenthesis space equals space 3 straight e to the power of 2 x end exponent space plus space 1 for x element of straight real numbers
straight g left parenthesis x right parenthesis space equals space x space plus space 1 for x element of straight real numbers

(i)
Write down the range of straight f and the range of straight g.

(ii)
Findspace straight g squared left parenthesis 0 right parenthesis space.

(iii)
Hence find fg squared left parenthesis 0 right parenthesis .

(iv)
On the axes below, sketch the graphs of y space equals space straight f left parenthesis x right parenthesis and y space equals space straight f to the power of space – 1 end exponent left parenthesis x right parenthesis .
State the intercepts with the coordinate axes and the equations of any asymptotes.
q6a-2025-specimen-paper-1-cie-igcse-additional-maths

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2a
Sme Calculator3 marks

The functionspace straight f is defined byspace straight f left parenthesis x right parenthesis space equals space ln left parenthesis 2 x plus 1 right parenthesis space for space x space greater or equal than space 0.

Sketch the graph of y space equals space straight f left parenthesis x right parenthesis and hence sketch the graph of y space equals space straight f to the power of negative 1 end exponent space left parenthesis x right parenthesis on the axes below.

q11-0606-s20-qp-21-additional-maths

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2b
Sme Calculator7 marks

The function straight g is defined by straight g left parenthesis x right parenthesis space equals space left parenthesis x minus 4 right parenthesis squared space plus 1 space for space x space less or equal than space 4.

(i)
Find an expression for straight g to the power of negative 1 end exponent space left parenthesis x right parenthesis and state its domain and range.

[4]

(ii)
Find and simplify an expression for fg left parenthesis x right parenthesis.

[2]

(iii)
Explain why the function gf does not exist.

[1]

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3a
Sme Calculator2 marks

straight f left parenthesis x right parenthesis space equals space 3 plus straight e to the power of x space for space x space element of space straight real numbers
straight g left parenthesis x right parenthesis space equals space 9 x minus 5 space for space x space element of space straight real numbers

Find the range of straight f and of straight g.

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3b
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Find the exact solution of straight f to the power of negative 1 end exponent space left parenthesis x right parenthesis space equals space straight g apostrophe left parenthesis x right parenthesis .

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3c
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Find the solution of straight g squared space left parenthesis x right parenthesis space equals space 112.

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