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Example
Paper 1 – Composite and Inverse Functions
2
Let f ( x) log x � .
2
� x , for 0
( a ) Find f
1 ( x) .
�
( b ) Write down the range of
Let g( x) log x x � .
3
� , for 0
1
f � .
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( c ) Find the value of
�1
( f g)( 100 ) , giving your answer as an integer .
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Solution
( a )
y� log x
2
2
� � ( M1 ) for swapping variables x
2
10 x � y
log y
10 x � y 1 x ( 10 ) 2
� y ( A1 ) for changing subject
1 �1 2
� f ( x) � 10 x A1
( b ) { y: y� 0 } A1
( c )
3 g( 100 ) � log100 ( M1 ) for substitution g( 100 ) � 6 ( A1 ) for correct value
1 1
( f �
� g)( 100 ) � f ( g( 100 ))
1 1
( f �
� g)( 100 ) � f ( 6 )
1 ( 6 ) �1 2
( f g)( 100 ) � 10 ( M1 ) for substitution
�1
( f g)( 100 ) 1000
� A1
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[ 1 ]
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