Question

Let A = {−1, 0, 1, 2}, B = {−4, −2, 0, 2} and f , g : A → B be functions defined by f ( x ) = x 2 − x , x ∈ A and . Are f and g equal? Justify your answer. (Hint: One may note that two function f : A → B and g: A → B such that f ( a ) = g( a ) &mnForE; a ∈ A , are called equal functions).

Answer 1

Given that f,g:A→B are the functions defined by,

f( x )= x 2 −x,x∈A g( x )=2| x− 1 2 |−1,x∈A

And A={ −1,0,1,2 },B={ −4,−2,0,2 }

Calculate f( −1 ),

f( −1 )= ( −1 ) 2 −( −1 ) f( −1 )=2

Calculate g( −1 ),

g( −1 )=2| ( −1 )− 1 2 |−1 g( −1 )=2( 3 2 )−1 g( −1 )=2

It can be observed that f( −1 )=g( −1 ).

Calculate f( 0 ),

f( 0 )= 0 2 −( 0 ) f( 0 )=0

Calculate g( 0 ),

g( 0 )=2| ( 0 )− 1 2 |−1 g( 0 )=2( 1 2 )−1 g( 0 )=0

It can be observed that f( 0 )=g( 0 ).

Calculate f( 1 ),

f( 1 )= ( 1 ) 2 −( 1 ) f( 1 )=0

Calculate g( 1 ),

g( 1 )=2| ( 1 )− 1 2 |−1 g( 1 )=2( 1 2 )−1 g( 1 )=0

It can be observed that f( 1 )=g( 1 ).

Calculate f( 2 ),

f( 2 )= ( 2 ) 2 −( 2 ) f( 2 )=2

Calculate g( 2 ),

g( 2 )=2| ( 2 )− 1 2 |−1 g( 2 )=2( 3 2 )−1 g( 2 )=2

It can be observed that f( 2 )=g( 2 ).

So, f( a )=g( a )   for all a∈A

Hence, the functions f and g are equal.