Let A = x [ ] R : x is not a positive integer.

Question

Let A = {x $$ \in $$ R : x is not a positive integer}.

Define a function $f$ : A $$ \to $$  R   as  $f(x)$ = $${{2x} \over {x - 1}}$$,

then $f$ is :

Accepted Answer

f(x) = $${{2x} \over {x - 1}}$$

f(x) = 2 + $${2 \over {x - 1}}$$

f'(x) = - $${2 \over {{{\left( {x - 1} \right)}^2}}}$$ < 0 $$\forall $$ x $$ \in $$ R

Hence f(x) is strictly decreasing

So, f(x) is one-one

Range :  Let y = $${{2x} \over {x - 1}}$$

xy - y = 2x

$$ \Rightarrow $$  x(y - 2) = y

$$ \Rightarrow $$  x = $${y \over {y - 2}}$$

given that x $$ \in $$ R  : x is not a +ve integer

$$ \therefore $$  $${y \over {y - 2}} \ne $$ N    (N $$ \to $$ Natural number)

$$ \Rightarrow $$  y $$ \ne $$ Ny - 2N

$$ \Rightarrow $$  y $$ \ne $$ $${{2N} \over {N - 1}}$$

So range $$ \notin $$ R (in to function)

Solutions · 1