IGCSEMaths: ExtendedCIEExam Questions2. Algebra & Graphs2.10 Functions

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First teaching 2021

Last exams 2024

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Functions (CIE IGCSE Maths: Extended)

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2.10 Functions

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2.10 Functions

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1a
Sme Calculator2 marks

straight f open parentheses x close parentheses space equals space x to the power of 2 space end exponent plus space 1 space space space space space space space space space space space straight g open parentheses x close parentheses space equals space 1 minus space 2 x space space space space space space space space space space space straight h open parentheses x close parentheses space equals space 1 over x comma space x space not equal to 0 space space space space space space space space space space straight j open parentheses x close parentheses space equals space 5 to the power of x 

Find the value of 

i)
straight f open parentheses 3 close parentheses,
 
[1]
ii)
gf open parentheses 3 close parentheses.
 
[1]

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1b
Sme Calculator2 marks

Find straight g to the power of negative 1 end exponent space open parentheses x close parentheses.

straight g to the power of negative 1 end exponent space open parentheses x close parentheses space equals space ................................................ 

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1c
Sme Calculator1 mark

Find x when straight h left parenthesis x right parenthesis space equals space 2.


x space equals ................................................ 

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2a
Sme Calculator1 mark

straight f open parentheses x close parentheses equals 7 x minus 4      straight g open parentheses x close parentheses equals fraction numerator 2 x over denominator x minus 3 end fraction comma space x not equal to 3   straight h open parentheses x close parentheses equals x squared

Find space straight g open parentheses 6 close parentheses.

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2b
Sme Calculator2 marks

Find space fg open parentheses 4 close parentheses.

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2c
Sme Calculator1 mark

Find space fh open parentheses x close parentheses.

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3a
Sme Calculator2 marks

straight f left parenthesis x right parenthesis space equals space 2 x space plus space 1       straight g left parenthesis x right parenthesis space equals space x squared space plus space 4        

Solve the equation straight f left parenthesis x right parenthesis space equals space straight g left parenthesis 1 right parenthesis.

x space equals .............................................. 

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3b
Sme Calculator2 marks

Find straight f to the power of negative 1 end exponent left parenthesis x right parenthesis.


 straight f to the power of negative 1 end exponent left parenthesis x right parenthesis space equals .............................................. 

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3c
Sme Calculator3 marks

Find gf left parenthesis x right parenthesis in its simplest form.

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4
Sme Calculator5 marks

straight f left parenthesis x right parenthesis equals 7 minus 2 x space space space space space space space space space space space space space space space straight g left parenthesis x right parenthesis equals 10 over x comma space x not equal to 0 space space space space space space space space space space space space space space space straight h left parenthesis x right parenthesis equals 27 to the power of x

Find

i)
straight f left parenthesis negative 3 right parenthesis,

 

[1]

ii)
hg left parenthesis 30 right parenthesis,

 

[2]

iii)
straight f to the power of negative 1 end exponent left parenthesis x right parenthesis.

 

straight f to the power of negative 1 end exponent left parenthesis x right parenthesis equals................................................... [2]

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5a
Sme Calculator6 marks

straight f open parentheses x close parentheses equals 2 x minus 3 space space space space space space space space space space space space space space space straight g open parentheses x close parentheses equals 9 minus x squared space space space space space space space space space space space space space space space straight h open parentheses x close parentheses equals 3 to the power of x

Find

i)

straight f open parentheses 4 close parentheses,
 

[1]

ii)

hg open parentheses 3 close parentheses,
 

[2]

iii)

straight g open parentheses 2 x close parentheses in its simplest form,
 

[1]

iv)

fg open parentheses x close parentheses, in its simplest form.
 

[2]

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5b
Sme Calculator2 marks

Find straight f to the power of negative 1 end exponent open parentheses x close parentheses.

straight f to the power of negative 1 end exponent open parentheses x close parentheses = ................................................... 

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6a
Sme Calculator2 marks
text f end text left parenthesis x right parenthesis equals 7 x minus 2 text g end text left parenthesis x right parenthesis equals x squared plus 1 text h end text left parenthesis x right parenthesis equals 3 to the power of x

Find text gh end text left parenthesis 2 right parenthesis.

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6b
Sme Calculator2 marks

Find text f end text to the power of negative 1 end exponent left parenthesis x right parenthesis.
 

text f end text to the power of negative 1 end exponent left parenthesis x right parenthesis = .................................................... 

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7a
Sme Calculator1 mark

straight f left parenthesis x right parenthesis equals fraction numerator 3 over denominator x plus 2 end fraction comma space x not equal to negative 2         straight g left parenthesis x right parenthesis equals 8 x minus 5         

Work out space straight g open parentheses 1 fourth close parentheses.

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7b
Sme Calculator2 marks

Work out space ff left parenthesis 2 right parenthesis.

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7c
Sme Calculator2 marks

Find space gg left parenthesis x right parenthesis, giving your answer in its simplest form.

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7d
Sme Calculator2 marks

Find space straight g to the power of negative 1 end exponent left parenthesis x right parenthesis.
 

straight g to the power of negative 1 end exponent left parenthesis x right parenthesis equals..................................................

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8a
Sme Calculator1 mark

straight f left parenthesis x right parenthesis equals 3 x plus 4 space space space space space space space space space space straight g left parenthesis x right parenthesis equals 2 x minus 1 space space space space space space space space space space straight h left parenthesis x right parenthesis equals 3 to the power of x

Find  straight g open parentheses 1 half close parentheses.

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8b
Sme Calculator2 marks

Find  fh left parenthesis negative 1 right parenthesis.

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8c
Sme Calculator2 marks

Find  straight g to the power of negative 1 end exponent left parenthesis x right parenthesis.
 

straight g to the power of negative 1 end exponent left parenthesis x right parenthesis equals ................................................. 

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8d
Sme Calculator2 marks

Find  ff left parenthesis x right parenthesis in its simplest form.

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9
Sme Calculator8 marks

straight f left parenthesis x right parenthesis equals 2 x minus 3 space space space space space space space space space space straight g left parenthesis x right parenthesis equals x squared plus 1

i)
Find gg left parenthesis 2 right parenthesis.

[2]
ii)
Find  straight g left parenthesis x plus 2 right parenthesis, giving your answer in its simplest form.

[2]
iii)
Find x when straight f left parenthesis x right parenthesis equals 7.

x equals............................................... [2]
iv)
Find straight f to the power of negative 1 end exponent left parenthesis x right parenthesis.

straight f to the power of negative 1 end exponent left parenthesis x right parenthesis equals............................................... [2]

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1a
Sme Calculator1 mark

space space space space space space space space space space straight g open parentheses x close parentheses space equals space 1 minus space 2 x space space space space space space space space space space space straight h open parentheses x close parentheses space equals space 1 over x comma space x space not equal to 0 space space space space space space space space space space straight j open parentheses x close parentheses space equals space 5 to the power of x

 

Find hh left parenthesis x right parenthesis, giving your answer in its simplest form.

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1b
Sme Calculator1 mark

Find straight j open parentheses 5 close parentheses.

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1c
Sme Calculator2 marks

   

Find the value of x.

x space equals ................................................ 

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2a
Sme Calculator2 marks

straight f open parentheses x close parentheses space equals space 4 space minus space 3 x     straight g open parentheses x close parentheses space equals space x to the power of 2 space end exponent plus space x      straight h open parentheses x close parentheses space equals space 3 to the power of x

Find  fh open parentheses 2 close parentheses.

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2b
Sme Calculator2 marks

Find  straight f to the power of negative 1 end exponent open parentheses x close parentheses.

 

straight f to the power of negative 1 end exponent open parentheses x close parentheses space equals space .................................................

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2c
Sme Calculator6 marks

Simplify.

i)
straight f open parentheses 1 space minus space 2 x close parentheses
 
[2]
ii)
gf open parentheses x close parentheses space minus space 9 straight g open parentheses x close parentheses
 
[4]

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3a
Sme Calculator1 mark

straight f open parentheses x close parentheses equals 3 x plus 2      straight g open parentheses x close parentheses equals x squared plus 1      straight h open parentheses x close parentheses equals 4 to the power of x

Find space straight h open parentheses 4 close parentheses.

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3b
Sme Calculator2 marks

Find space fg open parentheses 1 close parentheses.

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3c
Sme Calculator3 marks

Find space gf open parentheses x close parentheses in the form space a x squared plus b x plus c.

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3d
Sme Calculator2 marks

Find x when space straight f open parentheses x close parentheses equals straight g open parentheses 7 close parentheses.
 

x equals................................................ 

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3e
Sme Calculator2 marks

Find space straight f to the power of negative 1 end exponent open parentheses x close parentheses.
 

straight f to the power of negative 1 end exponent open parentheses x close parentheses equals................................................ 

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4a
Sme Calculator2 marks

space straight f left parenthesis x right parenthesis equals 4 x minus 1        space straight g left parenthesis x right parenthesis equals x squared        space straight h left parenthesis x right parenthesis equals 3 to the power of negative x end exponent

Find in its simplest form

i)
space straight f left parenthesis x minus 3 right parenthesis,


[1]

ii)
space straight g left parenthesis 5 x right parenthesis.


[1]

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4b
Sme Calculator2 marks

Findspace straight f to the power of negative 1 end exponent left parenthesis x right parenthesis.


space straight f to the power of negative 1 end exponent left parenthesis x right parenthesis equals ................................................. 

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4c
Sme Calculator3 marks

Find the value ofspace hh left parenthesis 1 right parenthesis, correct to 4 significant figures.

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5
Sme Calculator3 marks

straight g left parenthesis x right parenthesis equals 10 over x comma space x not equal to 0 space.
 

Solve.

straight g left parenthesis 2 x plus 1 right parenthesis equals 4

x = ..................................................

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6a
Sme Calculator2 marks

straight f open parentheses x close parentheses equals 2 x minus 3 space space space space space space space space space space space space space space space straight g open parentheses x close parentheses equals 9 minus x squared space space space space
 

Find x when  5 straight f open parentheses x close parentheses equals 3.

x = ................................................... 

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6b
Sme Calculator4 marks

Solve the equation  gf open parentheses x close parentheses equals negative 16.

x = .................... or x = .................... 

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7
Sme Calculator7 marks

straight g left parenthesis x right parenthesis equals 8 x minus 5 space space space space space space space space space space space space space space straight h left parenthesis x right parenthesis equals x squared plus 6

i)
Show that space hg left parenthesis x right parenthesis equals 19 space simplifies to space 16 x squared minus 20 x plus 3 equals 0.
 

[3]

ii)
Use the quadratic formula to solve space 16 x squared minus 20 x plus 3 equals 0.
Show all your working and give your answers correct to 2 decimal places.
 

x equals......................... or x equals......................... [4]

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8
Sme Calculator2 marks

straight f left parenthesis x right parenthesis equals 3 x plus 4.
 

Find  open parentheses straight f left parenthesis x right parenthesis close parentheses squared  in the form  a x squared plus b x plus c.

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9
Sme Calculator8 marks

straight f left parenthesis x right parenthesis equals 8 minus 3 x        straight g left parenthesis x right parenthesis equals fraction numerator 10 over denominator x plus 1 end fraction comma space x not equal to negative 1         straight h left parenthesis x right parenthesis equals 2 to the power of x

Find

i)table row blank row blank end table
hf open parentheses 8 over 3 close parentheses space,

[2]
ii)
gh left parenthesis negative 2 right parenthesis, 

[2]
iii)
straight g to the power of negative 1 end exponent left parenthesis x right parenthesis,

straight g to the power of negative 1 end exponent left parenthesis x right parenthesis equals ................................................ [3]
iv)
straight f to the power of negative 1 end exponent straight f left parenthesis 5 right parenthesis.

[1]

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10
Sme Calculator2 marks

straight f open parentheses x close parentheses equals 7 x minus 4      

Find the value of x when space straight f open parentheses x plus 2 close parentheses equals negative 11.
 

x equals................................................ 

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1
Sme Calculator4 marks

straight g open parentheses x close parentheses space equals space 1 minus space 2 x 

Find the value of 

Find straight g left parenthesis x right parenthesis straight g left parenthesis x right parenthesis space minus space gg left parenthesis x right parenthesis, giving your answer in the form a x to the power of 2 space end exponent plus space b x space plus space c.

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2
Sme Calculator2 marks

straight h open parentheses x close parentheses space equals space 3 to the power of x.

Find the value of k for which fraction numerator 1 over denominator straight h open parentheses x close parentheses end fraction space equals space 9 to the power of k x end exponent

     
k space equals ................................................ 

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3
Sme Calculator3 marks

straight f open parentheses x close parentheses equals 3 x plus 2            straight g open parentheses x close parentheses equals x squared plus 1

Find space fraction numerator straight g open parentheses x close parentheses over denominator straight f open parentheses x close parentheses end fraction plus x.
Give your answer as a single fraction, in terms of x, in its simplest form.

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4a
Sme Calculator2 marks

space straight f left parenthesis x right parenthesis equals 4 x minus 1        space straight g left parenthesis x right parenthesis equals x squared        space straight h left parenthesis x right parenthesis equals 3 to the power of negative x end exponent

Show thatspace straight g left parenthesis 3 x minus 2 right parenthesis minus straight h left parenthesis negative 3 right parenthesis can be written as 9 x squared minus 12 x minus 23.

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4b
Sme Calculator2 marks

Find x whenspace straight f left parenthesis 61 right parenthesis equals straight h left parenthesis x right parenthesis.


x equals ................................................. 

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5a
Sme Calculator1 mark

straight h left parenthesis x right parenthesis space equals space 2 to the power of x

Solve the equation space h to the power of negative 1 end exponent left parenthesis x right parenthesis space equals space 0.5.


x space equals .............................................. 

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5b
Sme Calculator1 mark

fraction numerator 1 over denominator straight h left parenthesis x right parenthesis end fraction space equals space 2 to the power of k x end exponent

Write down the value of k.


k space equals .............................................. 

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6a
Sme Calculator3 marks

straight f left parenthesis x right parenthesis equals 7 minus 2 x space space space space space space space space space space space space space space space straight g left parenthesis x right parenthesis equals 10 over x comma space x not equal to 0 space space space space space space space space space space space space space space space straight h left parenthesis x right parenthesis equals 27 to the power of x

Simplify, giving your answer as a single fraction.

fraction numerator 1 over denominator straight f left parenthesis x right parenthesis end fraction plus straight g left parenthesis x right parenthesis

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6b
Sme Calculator1 mark

Find  straight h to the power of negative 1 end exponent left parenthesis 19 space 683 right parenthesis.

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7
Sme Calculator1 mark

space straight h open parentheses x close parentheses equals 3 to the power of x.

Find x when text h end text to the power of negative 1 end exponent open parentheses x close parentheses equals negative 2.

x = ................................................... 

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8a
Sme Calculator3 marks
text f end text left parenthesis x right parenthesis equals 7 x minus 2 text g end text left parenthesis x right parenthesis equals x squared plus 1 text h end text left parenthesis x right parenthesis equals 3 to the power of x

text gg end text left parenthesis x right parenthesis equals a x to the power of 4 plus b x squared plus c 
 
Find the values of a, b, and c. 
 

a = ....................................................
b = ....................................................
c = ....................................................

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8b
Sme Calculator3 marks

Find x when text hf end text left parenthesis x right parenthesis equals 81. 
 

x = .................................................... 

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9
Sme Calculator3 marks

The diagram shows the graph of y equals text f end text left parenthesis x right parenthesis where text f end text left parenthesis x right parenthesis equals x squared minus 2 over x minus 2 comma space x not equal to 0.

cie-igcse-2019-may-jun-p4-tz3-q5a

Use the graph to find

i)
text f end text left parenthesis 1 right parenthesis.
 
[1]
ii)
text ff end text left parenthesis negative 2 right parenthesis.
 
[2]

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10
Sme Calculator2 marks

straight g open parentheses x close parentheses equals 2 x minus 1 space space space space space space space space space space space space space space space space space space space straight h open parentheses x close parentheses equals 3 to the power of x

Find  x when  h to the power of negative 1 end exponent open parentheses x close parentheses equals g left parenthesis 2 right parenthesis.
 

x = ................................................. 

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11
Sme Calculator3 marks

straight h left parenthesis x right parenthesis equals x to the power of x comma space x greater than 0

i)
Calculate straight h left parenthesis 0.3 right parenthesis.
Give your answer correct to 2 decimal places.

[2]
ii)

Find x when straight h left parenthesis x right parenthesis equals 256.

x equals............................................... [1]

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12
Sme Calculator1 mark

space space straight j open parentheses x close parentheses space equals space 5 to the power of x.

Find space x spacewhen space straight j to the power of negative 1 end exponent open parentheses x close parentheses space equals space 2.

x space equals space................................................  

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13
Sme Calculator1 mark

straight h open parentheses x close parentheses equals 4 to the power of x.

Find x when space straight h to the power of negative 1 end exponent open parentheses x close parentheses equals 2.
 

x equals................................................ 

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14
Sme Calculator2 marks

straight h open parentheses x close parentheses equals x squared

Find the values of space p that satisfy space straight h open parentheses p close parentheses equals p.

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