Question

( 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

# 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 +

Solution