Question

A function $f:(R-A)\to R$ is defined as $f\left(x\right)=\frac{x+2}{x^2-3x+4},$ 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)=___.

Answer 1

$f\left(x\right)=\frac{x+2}{x^2-3x+4}=\frac{Nr\left(x\right)}{Dr\left(x\right)}$
Since f(x) is a rational funtion, it is defined at all the points where the denominator is non-zero.
$Dr\left(x\right)=x^2-3x+4=x^2-3x+\frac{9}{4}+\frac{7}{4}={(x-\frac{3}{2})}^2+\frac{7}{4}$
Now, ${(x-\frac{3}{2})}^2\ge 0\Rightarrow {(x-\frac{3}{2})}^2+\frac{7}{4}\ge \frac{7}{4}$
Hence, the denominator is always positive and hence non-zero.
f(x) is defined for all $xϵR.$
$\Rightarrow A=\varphi$
$\Rightarrow n\left(A\right)=0$