Question

16. Show that the function f: R + R given by f (x) = x3 + x is a bijective function 2x – 1 12 Iff, R. Rio defined bufm- mnono +

11th - 12th Class

Maths

Solution

418

4.0 (1 ratings)

( f: R rightarrow R quad f(x)=x^{3}+x ) Let ( x, y ) belongs to ( R ) ( f(x)=f(y) ) ( x^{3}+x=y^{3}+y ) ( left(x^{3}-y^{3}right)+x-y=0 ) ( (x-y)left(x^{2}+x y+y^{2}+1right)=0 ) ( x=y ) ( f(x)=f(y) ) ( =) x=y ) for all ( x y ) belongs to ( R ) So, ( mathrm{f} ) is one-one function Let y be any arbitrary element of R ( f(x)=y ) ( =x^{3}+x=y ) ( x^{3}+x-y=0 ) For every value of y, the equation ( x^{3}+x-y=0 ) has a real root a (alpha) ( a^{3}+a-y=0 ) ( a^{3}+a=y ) ( f(a)=y ) For every y belongs to R there exists a (alpha) belongs R such that ( f(a)=y ) So, fis a onto function Hence, ( mathrm{f}: mathrm{R} rightarrow mathrm{R} ) is a bijective function