if f:(1,infinity)- [1,infinity) defined by f(x)=x+2/x-1 then 1) f is one-one onto 2) f is one-one - Maths - Relations and Functions

Question

if f:(1,infinity)-> [1,infinity) defined by f(x)=x+2/x-1 then 1)
f is one-one & onto 2) f is one-one but not onto 3) f is not
one-one but onto 4) f is neither one-one nor onto

Ajanta Trivedi · Accepted Answer

if the given function is
f:(1,∞)→[1,∞) ; f(x)=x+2x-1
to check that function is one -one,
let us consider x1 , x2∈(1,∞)then let us assume that f(x1)=f(x2)x1+2x1-1=x2+2x2-1⇒x1x2+2x2-x1-2=x1x2-x2+2x1-2⇒3x2=3x1⇒x1=x2therefore f(x) is one -one
now let y=x+2x-1xy-y=x+2x(y-1)=y+2x=y+2y-1=y-1+3y-1x=1+3y-1which is defined for all values of y in (1,∞)but not defined for 1
thus f(x) does not have the pre-image of 1thus f(x) is not onto
therefore function is one - one but not onto

hope this helps you

if f:(1,infinity)-> [1,infinity) defined by f(x)=x+2/x-1 then 1) f is one-one & onto. 2) f is one-one but not onto. 3) f is not one-one but onto 4) f is neither one-one nor onto