Which statement(s) verify that f ( x ) = 9 x − 1 f ( x ) = 9 x - 1 f ( x ) = 9 x − 1 and g ( x ) = 1 9 ( x + 1 ) g ( x ) = \frac { 1 } { 9 } ( x + 1 ) g ( x ) = 9 1 ( x + 1 ) are inverse?A) f ( x ) ⋅ g ( x ) = ( 9 x − 1 ) 1 9 ( x + 1 ) = − 1 f ( x ) \cdot g ( x ) = ( 9 x - 1 ) \frac { 1 } { 9 } ( x + 1 ) = - 1 f ( x ) ⋅ g ( x ) = ( 9 x − 1 ) 9 1 ( x + 1 ) = − 1 B) f ( g ( x ) ) = ( 9 ( 1 9 x ) + 1 ) − 1 = x , g ( f ( x ) ) = 1 9 ( 9 ( x − 1 ) + 1 ) = x f ( g ( x ) ) = \left( 9 \left( \frac { 1 } { 9 } x \right) + 1 \right) - 1 = x , g ( f ( x ) ) = \frac { 1 } { 9 } ( 9 ( x - 1 ) + 1 ) = x f ( g ( x ) ) = ( 9 ( 9 1 x ) + 1 ) − 1 = x , g ( f ( x ) ) = 9 1 ( 9 ( x − 1 ) + 1 ) = x C) f ( g ( x ) ) = ( 9 ( 1 9 x ) + 1 ) − 1 = x ; g ( f ( x ) ) = 1 9 ( ( 9 x − 1 ) + 1 ) = x f ( g ( x ) ) = \left( 9 \left( \frac { 1 } { 9 } x \right) + 1 \right) - 1 = x ; g ( f ( x ) ) = \frac { 1 } { 9 } ( ( 9 x - 1 ) + 1 ) = x f ( g ( x ) ) = ( 9 ( 9 1 x ) + 1 ) − 1 = x ; g ( f ( x ) ) = 9 1 (( 9 x − 1 ) + 1 ) = x D) f ( g ( x ) ) = 9 ( 1 9 ( x + 1 ) ) − 1 = x ; g ( f ( x ) ) = 1 9 ( ( 9 x − 1 ) + 1 ) = x f ( g ( x ) ) = 9 \left( \frac { 1 } { 9 } ( x + 1 ) \right) - 1 = x ; g ( f ( x ) ) = \frac { 1 } { 9 } ( ( 9 x - 1 ) + 1 ) = x f ( g ( x ) ) = 9 ( 9 1 ( x + 1 ) ) − 1 = x ; g ( f ( x ) ) = 9 1 (( 9 x − 1 ) + 1 ) = x E) f ( x ) g ( x ) = 9 ( x − 1 ) 9 ( x + 1 ) = − 1 \frac { f ( x ) } { g ( x ) } = \frac { 9 ( x - 1 ) } { 9 ( x + 1 ) } = - 1 g ( x ) f ( x ) = 9 ( x + 1 ) 9 ( x − 1 ) = − 1

Question

Which statement(s) verify that   f ( x )  = 9 x − 1 f ( x )  = 9 x - 1 f ( x )  = 9 x − 1   and   g ( x )  = 1 9 ( x + 1 )  g ( x )  = \frac { 1 } { 9 } ( x + 1 )  g ( x )  = 9 1 ​ ( x + 1 )    are inverse?A)    f ( x )  ⋅ g ( x )  = ( 9 x − 1 )  1 9 ( x + 1 )  = − 1 f ( x )  \cdot g ( x )  = ( 9 x - 1 )  \frac { 1 } { 9 } ( x + 1 )  = - 1 f ( x )  ⋅ g ( x )  = ( 9 x − 1 )  9 1 ​ ( x + 1 )  = − 1 B)    f ( g ( x )  )  = ( 9 ( 1 9 x )  + 1 )  − 1 = x , g ( f ( x )  )  = 1 9 ( 9 ( x − 1 )  + 1 )  = x f ( g ( x )  )  = \left( 9 \left( \frac { 1 } { 9 } x \right)  + 1 \right)  - 1 = x , g ( f ( x )  )  = \frac { 1 } { 9 } ( 9 ( x - 1 )  + 1 )  = x f ( g ( x ) )  = ( 9 ( 9 1 ​ x )  + 1 )  − 1 = x , g ( f ( x ) )  = 9 1 ​ ( 9 ( x − 1 )  + 1 )  = x C)    f ( g ( x )  )  = ( 9 ( 1 9 x )  + 1 )  − 1 = x ; g ( f ( x )  )  = 1 9 ( ( 9 x − 1 )  + 1 )  = x f ( g ( x )  )  = \left( 9 \left( \frac { 1 } { 9 } x \right)  + 1 \right)  - 1 = x ; g ( f ( x )  )  = \frac { 1 } { 9 } ( ( 9 x - 1 )  + 1 )  = x f ( g ( x ) )  = ( 9 ( 9 1 ​ x )  + 1 )  − 1 = x ; g ( f ( x ) )  = 9 1 ​ (( 9 x − 1 )  + 1 )  = x D)    f ( g ( x )  )  = 9 ( 1 9 ( x + 1 )  )  − 1 = x ; g ( f ( x )  )  = 1 9 ( ( 9 x − 1 )  + 1 )  = x f ( g ( x )  )  = 9 \left( \frac { 1 } { 9 } ( x + 1 )  \right)  - 1 = x ; g ( f ( x )  )  = \frac { 1 } { 9 } ( ( 9 x - 1 )  + 1 )  = x f ( g ( x ) )  = 9 ( 9 1 ​ ( x + 1 )  )  − 1 = x ; g ( f ( x ) )  = 9 1 ​ (( 9 x − 1 )  + 1 )  = x E)    f ( x )  g ( x )  = 9 ( x − 1 )  9 ( x + 1 )  = − 1 \frac { f ( x )  } { g ( x )  } = \frac { 9 ( x - 1 )  } { 9 ( x + 1 )  } = - 1 g ( x )  f ( x )  ​ = 9 ( x + 1 )  9 ( x − 1 )  ​ = − 1

pedro martinez · Accepted Answer

Final Answer : D, Explanation :   Two functions are inverses if and only if   f ( g ( x ) ) = x f(g(x)) = x f ( g ( x )) = x   and   g ( f ( x ) ) = x g(f(x)) = x g ( f ( x )) = x  . Choice D correctly applies these operations and simplifies to show that each composition equals   x x x  , verifying that   f f f   and   g g g   are inverse functions.

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