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  4. The range of the function f(x)=(x2+2/x2+1), x ∈ R , is

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

J & K CETJ & K CET 2016Relations and Functions

Solution:

Let $f(x) = \frac{x^2 +2}{x^2 +1} = y$
$\Rightarrow x^2y + y = x^2 +2$
$ \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$
$\therefore R_f = (1, 2]$