Question

Let a function $f:R\to R$ be defined as $f(x)=\{\begin{array}{ll}\sin x-e^x& \text{ if }x\le 0\\ a+[-x]& \text{ if }0<x<1\\ 2x-b& \text{ if }x\ge 1\end{array}$
Where $[x]$ is the greatest integer less than or equal to $x$. If $f$ is continuous on $R$, then $(a+b)$ is equal to:

• A. 4
• B. 3
• C. 2
• D. 5

Answer 1

Answer: (B)

Continuous at $x=0$
$f\left(0^{+}\right)=f\left(0^{-}\right)\Rightarrow a-1=0-e^0$
$\Rightarrow a=0$
Continuous at $x=1$
$f\left(1^{+}\right)=f\left(1^{-}\right)$
$\Rightarrow 2(1)-b=a+(-1)$
$\Rightarrow b=2-a+1\Rightarrow b=3$
$\therefore a+b=3$