Question

If $f\left(x\right)=\{\begin{array}{cc}{x}^2+2& x\ge 2\\ 1-x& x<2\end{array}$ and $g\left(x\right)=\{\begin{array}{cc}2x& x>2\\ 3-x& x\ge 2\end{array}$. then f(g(x)) is

• A. Continuous at $x\in R-\left\{2\right\}$
• B. Continuous $x\in R-\left\{1\right\}$
• C. Continuous $x\in R$
• D. Continuous $x\in R-\{0,1,2\}$

Answer 1

Answer: (A)

Continuous at $x\in R-\left\{2\right\}$

Given $f\left(x\right)=\{\begin{array}{cc}{x}^2+2& x\ge 2\\ 1-x& x<2\end{array}$ and $g\left(x\right)=\{\begin{array}{cc}2x& x\ge 2\\ 3-x& x<2\end{array}$.

Now, $f\left(g\right(x\left)\right)=4x^2+2$ for $x\ge 2$ and

$=x-2$ for $x<2$

Now for continuity of $f\left(g\right(x\left)\right)$ at $x=2$:

$\underset{x\to 2^{+}}{lim}=\underset{x\to 2^{-}}{lim}$

But,

$f\left(g\right(2^{+}\left)\right)=18$ and $f\left(g\right(2^{-}\left)\right)=0$

So, $f\left(g\right(x\left)\right)$ is continuous at $x\in R-\left\{2\right\}$