Question

If y is a function of x given by $2\log y-\log x-\log (y-1)=0$ then

• A. domain $=[4,+\infty ),$ range $=(1,+\infty )$
• B. domain $=[4,+\infty ),$ range $=[2,+\infty )$
• C. domain $=[2,+\infty ),$ range $=(2,+\infty )$
• D. none of these

Answer 1

The correct option is B domain $=[4,+\infty ),$ range $=[2,+\infty )$

$2\log y-\log x-\log (y-1)=0$

Simplifying we get

$\frac{y^2}{y-1}=x\dots \left(1\right)$

$y^2=xy-x$

$y^2-xy+x=0$

$y=\frac{x\pm \sqrt{x^2-4x}}{2}$

Now $x^2-4x\ge 0$

$(x-2{)}^2-4\ge 0$

$(x-2{)}^2\ge 4$

$x-2\ge 2$ ...( since negative input of log is not possible, we donot consider another alternative $x-2<-4$)

$x\ge 4$

Hence domain of $f\left(x\right)=[4,\infty )$

From $\left(1\right)$,

$y-1>0$

Therefore range will be $[2,\infty )$