Let f = {(1, 1), (2, 3), (0, –1), (–1, –3)} be a function from Z to Z defined by f ( x ) = ax + b , for some integers a , b . Determine a , b .
Consider f = { ( 1 , 1 ) , ( 2 , 3 ) , ( 0 , − 1 ) , ( − 1 , − 3 ) } is a function of f : Z → Z f ( x ) = a x + b for all integers a , b
Let y = f ( x )
y = a x + b (1)
From the function f ( x ) substitute the value ( 1 , 1 ) equation (1),
1 = a × 1 + b 1= a + b (2)
From the function f ( x ) substitute the value ( 2 , 3 ) equation (1),
3 = a × 2 + b 3 = 2 a + b (3)
Subtract equation (2) from equation (3),
3 − 1 = ( 2 a + b ) − ( a + b ) 2 = 2 a + b − a − b 2 = a + 0 2 = a
Substitute the value of a = 2 in equation (2),
1 = 2 + b 1 − 2 = b − 1 = b
Therefore, the value of a is 2 and the value of b is − 1 .
Consider
Let
From the function
From the function
Subtract equation (2) from equation (3),
Substitute the value of
Therefore, the value of
