Question

The function $f\left(x\right)=\frac{1}{u^2+u-2}$, where $u=\frac{1}{x-1}$, is discontinuous at the points

• A. $x=-2,1,\frac{1}{2}$
• B. $x=\frac{1}{2},1,2$
• C. $x=1,0$
• D. none of these

Answer 1

The correct option is B $x=\frac{1}{2},1,2$

The function $u=\frac{1}{x-1}$suffers a discontinuity at the point $x=1$

The function $f\left(x\right)=\frac{1}{u^2+u-2}$ suffers a discontinuity at the points where $u^2+u-2=0$ i.e., $u=-2$ and $u=1.$

Using these value of $u,$ the corresponding values of $x$ are obtained by solving the equations

$-2=\frac{1}{x-1}$ and $1=\frac{1}{x-1}$ i.e., $x=\frac{1}{2}$ and $x=2$

Hence, the composite function is discontinuous at three points $x=\frac{1}{2},x=1$ and $x=2.$