Question

Let $f\left(x\right)={\log }_2(x^2-6x+8).$ Then

• A. domain of $f\left(x\right)$ is $R-[2,4]$
• B. domain of $f\left(x\right)$ is $(2,4)$
• C. range of $f\left(x\right)$ is $R$
• D. range of $f\left(x\right)$ is $R^{+}$

Answer 1

Answer: (A), (C)

range of $f\left(x\right)$ is $R$

Given : $f\left(x\right)={\log }_2(x^2-6x+8)={\log }_2\left[\right(x-2\left)\right(x-4\left)\right]$
For domain : $(x-2)(x-4)>0$
$\Rightarrow x\in (-\infty ,2)\cup (4,\infty )$

Let $y=x^2-6x+8=(x-3{)}^2-1$
$\Rightarrow y>-1$ for $x\in (-\infty ,2)\cup (4,\infty )$
But $y$ should be positive.
$\therefore y>0$
Since ${\log }_{a}x$ is an increasing function for $a>1,$
$\Rightarrow$ Range is $(-\infty ,\infty )$