Question

The range of the function $f\left(x\right)=\frac{x^2+2}{x^2+1},x\in R$ , is

• A. $(1,2]$
• B. $[1,2]$
• C. $(1,2)$
• D. $(1,5,2]$

Answer 1

Answer: (A)

Let $f(x)=\frac{x^2+2}{x^2+1}=y$
$\Rightarrow x^{2}y+y=x^2+2$
$\Rightarrow x=\pm \sqrt{\frac{2-y}{y-1}}$
Now, $x$ is defined if $2-y\ge 0$ and $y>1$
$\Rightarrow y\le 2$ and $y>1$
$\therefore R_f=(1,2]$