Question

Let a function \(f:\mathbb{R}\to\mathbb{R}\) be defined as
\(f(x)=\left\{\begin{matrix}\sin x-e^x&\text{if}&x\le0\\a+[-x]&\text{if}&0\lt x\lt1\\2x-b&\text{if}&x\ge1\end{matrix}\right.\)
where \([x]\) is the greatest integer less than or equal to \(x\). If \(f\) is continuous on \(\mathbb{R}\), then \((a+b)\) is equal to:

• A. 4
• B. 3
• C. 2
• D. 5

Hint:

For \(0\lt x\lt1\), \([-x]=-1\), so the middle branch is the constant \(a-1\); match it with the neighbouring branches at \(x=0\) and \(x=1\).

Answer 1

The Correct Option is B

For \(0\lt x\lt1\): \(-1\lt-x\lt0\Rightarrow[-x]=-1\), so \(f(x)=a-1\).
Continuity at \(x=0\): left value \(f(0)=\sin0-e^0=-1\); right limit \(=a-1\). So \(a-1=-1\Rightarrow a=0\).
Continuity at \(x=1\): left limit \(=a-1=-1\); \(f(1)=2-b\). So \(2-b=-1\Rightarrow b=3\).
\(a+b=0+3=3\).
Answer: (2) 3