Function f:R->R given by f(x)=x3+5 is function onto or not? because 0 belongs to R set ( codomain) but there does not exist x belong to R set (domain) such that f(x) =0 please give me detail explaination.

Asked by Kalyanrao Chavan | 19th Aug, 2014, 06:57: AM

Expert Answer:

For given function there exists value for x such that f(x)=0 which is given by
x cubed plus 5 equals 0 x cubed equals minus 5 x equals 3 root of minus 5 end root rightwards double arrow x equals minus 3 root of 5  S o space f o r space x equals minus 3 root of 5 comma space f left parenthesis x right parenthesis equals 0  O n e minus O n e space F u n c t i o n  L e t s space a s s u m e space f left parenthesis x subscript 1 right parenthesis equals f left parenthesis x subscript 2 right parenthesis comma space f o r space a n y space x subscript 1 space a n d space x subscript 2 space b e l o n g s space t o space R space left parenthesis D o m a i n right parenthesis f left parenthesis x subscript 1 right parenthesis equals f left parenthesis x subscript 2 right parenthesis x subscript 1 cubed plus 5 equals x subscript 2 cubed plus 5 x subscript 1 cubed equals x subscript 2 cubed  rightwards double arrow x subscript 1 equals x subscript 2 comma space H e n c e space f left parenthesis x right parenthesis space i s space o n e space o n e space f u n c t i o n  S i m i l a r l y space f o r space o n t o space f u n c t i o n comma space w e space k n o w space t h a t space a l l space v a l u e s space o f space c o d o m a i n space i s space a n space i m a g e space f o r space a t l e a s t space o n e space o f space t h e space e l e m e n t space i n space t h e space d o m a i n. space S o space R a n g e equals C o d o m a i n  x equals 3 root of y minus 5 end root   H e n c e space g i v e n space f u n c t i o n space i s space o n t o

Answered by Dharma Teja | 19th Aug, 2014, 07:38: AM