Question

What is the value of $a+b$ when function $f\left(x\right)$ given as

$f\left(x\right)=\left\{\begin{array}{l}\sin x-e^{x}x<0\\ a+\left|-x\right|0\le x<1\\ 2x-bx\ge 1\end{array}\right.$

is continuous in $(-\infty ,1]$.

Answer 1

Find the value of $a+b$.

As the function is continuous in $(-\infty ,1]$, so $\underset{x\to 1^{-}}{f\left(x\right)}=\underset{x\to 1^{+}}{f\left(x\right)}$.

For $x\to 1^{-}$, $f\left(x\right)=a+\left|-x\right|$ and for $x\to 1^{+}$, $f\left(x\right)=2x-b$. So the equation can be seen as:

$$\begin{gathered} \underset{x\to 1^{-}}{f\left(x\right)}=\underset{x\to 1^{+}}{f\left(x\right)} \\ \Rightarrow a+\left|-1\right|=2\left(1\right)-b \\ \Rightarrow a+1=2-b \\ \Rightarrow a+b=2-1 \\ \Rightarrow a+b=1 \end{gathered}$$

Thus the value of $a+b$ is $1$.

Hence, the correct answer is $1$.