A function $f : (R - A) \to R$ is defined as $f (x) = \frac{x + 2}{x^{2} - 3 x + 4},$ where R is one set of real numbers and A is a finite set of points where f(x) is not defined. Then n(A)=___.

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Solution

$f (x) = \frac{x + 2}{x^{2} - 3 x + 4} = \frac{N r (x)}{D r (x)}$
Since f(x) is a rational funtion, it is defined at all the points where the denominator is non-zero.
$D r (x) = x^{2} - 3 x + 4 = x^{2} - 3 x + \frac{9}{4} + \frac{7}{4} = {(x - \frac{3}{2})}^{2} + \frac{7}{4}$
Now, ${(x - \frac{3}{2})}^{2} \geq 0 \Rightarrow {(x - \frac{3}{2})}^{2} + \frac{7}{4} \geq \frac{7}{4}$
Hence, the denominator is always positive and hence non-zero.
f(x) is defined for all $x ϵ R .$
$\Rightarrow A = ϕ$
$\Rightarrow n (A) = 0$

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