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Functions (Cambridge (CIE) IGCSE International Maths: Extended): Exam Questions

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1a
Sme Calculator2 marks

straight f left parenthesis x right parenthesis space equals space 3 space – space 2 x space space space space straight g left parenthesis x right parenthesis space equals space 2 x space plus space 3

i) Find  space straight f left parenthesis – 3 right parenthesis.

[1]

ii) Find  gf left parenthesis – 3 right parenthesis.

[1]

1b
Sme Calculator2 marks

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

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

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2
Sme Calculator4 marks

straight g open parentheses x close parentheses space equals space 2 x space plus space 7         straight h open parentheses x close parentheses space equals space 3 x space minus space 8

i) Find gh left parenthesis x right parenthesis in its simplest form.

[2]

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

space straight g to the power of negative 1 end exponent open parentheses x close parentheses ................................................ [2]

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3a
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]

3b
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 ................................................ 

3c
Sme Calculator1 mark

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

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

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4a
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 .............................................. 

4b
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 .............................................. 

4c
Sme Calculator3 marks

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

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5
Sme Calculator2 marks

straight f left parenthesis x right parenthesis equals 3 x minus 5.  

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.................................................... 

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6a
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]

6b
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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7a
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.

7b
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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8
Sme Calculator2 marks

straight f left parenthesis x right parenthesis equals 7 plus 3 x.  

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 =  ................................................ 

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9
Sme Calculator3 marks

straight f left parenthesis x right parenthesis equals x cubed space space space space space space space space space space straight g left parenthesis x right parenthesis equals 5 x plus 2

i) Find  gf left parenthesis x right parenthesis.  

[1]

ii) 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 = ............................................... [2]

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10a
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.

10b
Sme Calculator2 marks

Find  fh left parenthesis negative 1 right parenthesis.

10c
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 ................................................. 

10d
Sme Calculator2 marks

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

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11
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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12a
Sme Calculator2 marks

      straight f open parentheses x close parentheses space equals space 3 x space minus space 2 straight g open parentheses x close parentheses space equals fraction numerator space 10 over denominator x plus 2 end fraction

Express the inverse function straight f to the power of negative 1 end exponent in the form straight f to the power of negative 1 end exponent open parentheses x close parentheses space equals space...

12b
Sme Calculator2 marks

Find  gf open parentheses x close parentheses
Simplify your answer.

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1a
Sme Calculator3 marks

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

Find x when gg left parenthesis x right parenthesis space equals space 7.

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

1b
Sme Calculator2 marks

Find x when space straight h to the power of negative 1 end exponent left parenthesis x right parenthesis space equals space 5 space.

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

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2
Sme Calculator2 marks

straight f open parentheses x close parentheses space equals space 3 x squared space plus space a  where space a space is an integer. straight f open parentheses negative 2 close parentheses space equals space 19

Find the value of a.

a space equals ................................................ 

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3a
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.

3b
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 .................................................

3c
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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4a
Sme Calculator2 marks

straight f left parenthesis x right parenthesis space equals space 4 x plus 3 space space space space space space space space space space space space space straight g left parenthesis x right parenthesis space equals space 5 x minus 4

fg left parenthesis x right parenthesis space equals space 20 x plus p

Find the value of p.

space p space equals .................................................. 

4b
Sme Calculator3 marks

straight h open parentheses x close parentheses equals fraction numerator 5 x minus 1 over denominator 3 end fraction

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

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

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5
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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6
Sme Calculator2 marks

f left parenthesis x right parenthesis equals 3 x minus 5      g left parenthesis x right parenthesis equals 2 to the power of x

Find f g left parenthesis 3 right parenthesis.

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7
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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8a
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 = ................................................... 

8b
Sme Calculator4 marks

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

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

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9
Sme Calculator2 marks

text f end text left parenthesis x right parenthesis equals 2 x plus 3  

Find text f end text left parenthesis 1 minus x right parenthesis in its simplest form.

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10
Sme Calculator2 marks

straight f left parenthesis x right parenthesis equals 7 plus 3 x space space space space space space space space space space space space space straight g left parenthesis x right parenthesis equals x to the power of 4  

Find the value of x when  straight f left parenthesis x right parenthesis equals straight g left parenthesis 2 right parenthesis.  

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

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11
Sme Calculator2 marks

straight h left parenthesis x right parenthesis equals a x squared plus 1

Find the value of a when straight h left parenthesis negative 2 right parenthesis equals 21.  

a = ............................................... 

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12
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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13a
Sme Calculator2 marks

text f end text left parenthesis x right parenthesis equals 5 minus 2 x         straight g left parenthesis x right parenthesis equals x squared plus 8

Calculate ff left parenthesis negative 3 right parenthesis

13b
Sme Calculator3 marks

Find 

i) straight g left parenthesis 2 x right parenthesis,

[1]

ii) 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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14
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) 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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15a
Sme Calculator2 marks

straight f open parentheses x close parentheses equals 7 minus x      straight g open parentheses x close parentheses equals 4 x plus 2      straight h open parentheses x close parentheses equals 15 minus x squared

Find ff open parentheses 2 close parentheses.

15b
Sme Calculator2 marks

Find gf open parentheses x close parentheses in its simplest form.

15c
Sme Calculator2 marks

Find straight h open parentheses 2 x close parentheses in its simplest form.

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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 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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3
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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4
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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5
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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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

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 = ....................................................

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 h left parenthesis x right parenthesis equals 3 to the power of x.

If  straight h left parenthesis 3 x right parenthesis equals k to the power of x, find the value of k.  

k = ................................................ 

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11
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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12
Sme Calculator5 marks

The functions straight f and straight g are such that

straight f left parenthesis x right parenthesis space equals space 5 x space plus space 3 space space space space straight g left parenthesis x right parenthesis space equals space a x space plus space b where a and b are constants.

straight g left parenthesis 3 right parenthesis space equals 20 space space space space space space space and space space space space space space straight f to the power of negative 1 end exponent left parenthesis 33 right parenthesis equals space straight g left parenthesis 1 right parenthesis

Find the value of a and the value of b.

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13
Sme Calculator4 marks

straight f open parentheses x close parentheses equals 1 half x      straight g open parentheses x close parentheses equals x minus x squared

Solve  straight f to the power of – 1 end exponent left parenthesis x right parenthesis space equals space gf left parenthesis x right parenthesis

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14
Sme Calculator5 marks

straight f open parentheses x close parentheses equals fraction numerator 2 x over denominator 5 end fraction minus 1

Work out the value of  straight f to the power of – 1 end exponent left parenthesis 3 right parenthesis space plus space straight f left parenthesis – 0.5 right parenthesis

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15
Sme Calculator3 marks

straight f open parentheses x close parentheses space equals space 5 minus x  and     straight g open parentheses x close parentheses space equals space 3 x plus 7

Simplify straight f left parenthesis 2 x right parenthesis space plus space straight g left parenthesis x space – space 1 right parenthesis

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