Question

If $f\left(x\right)=$$\{\begin{array}{c}|x|-3,x<1\\ |x-2|+a,x\ge 1\end{array},$ $g\left(x\right)=\{\begin{array}{c}2-|x|,x<2\\ \text{sgn}\left(x\right)-b,x\ge 2\end{array}$ and $h\left(x\right)=f\left(x\right)+g\left(x\right)$ is discontinuous at exactly one point, then which of the following values of $a$ and $b$ are possible

• A. $a=-3,b=0$
• B. $a=1,b=2$
• C. $a=2,b=1$
• D. $a=-3,b=2$

Answer 1

Answer: (A), (C), (D)

$a=-3,b=2$

$h\left(x\right)=f\left(x\right)+g\left(x\right)\Rightarrow h\left(x\right)=\{\begin{array}{cc}|x|-3+2-|x|,& x<1\\ |x-2|+a+2-|x|,& 1\le x<2\\ |x-2|+a+\text{sgn}\left(x\right)-b,& 2\le x<\infty \end{array}\Rightarrow h\left(x\right)=\{\begin{array}{cc}-1,& x\le 1\\ 4+a-2x,& 1\le x<2\\ x-1+a-b,& 2\le x<\infty \end{array}$

Now, for $h\left(x\right)$ discontinuity can occur at $x=1$ or $2$
At $x=1$ for $h\left(x\right)$ to be continuous
$\Rightarrow -1=4+a-2\Rightarrow a=-3$
At $x=2$ for $h\left(x\right)$ to be continuous
$\Rightarrow 4+a-4=2-1+a-b\Rightarrow b=1$
$h\left(x\right)$ will be continuous iff $a=-3$ and $b=1$
$\therefore$ For $h\left(x\right)$ to be discontinuous at exactly one point
$a=-3,b\ne 1$ or $a\ne -3,b=1$