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  • If, <img alt="f\left ( x \right )=-1+\left | x-2 \right |" src="http://entrancecorner.oncodecogs.com/gif.latex?f%5Cleft%20%28%20x%20%5Cright%20%29%3D-1&plus;%5Cleft%20%7C%20x-2%20%5Cright%

If, f\left ( x \right )=-1+\left | x-2 \right |  and; g\left ( x \right )=1-\left | x \right |  then the set of all points wherefog is discontinuous is :

Option: 1

\left \{ 0,2 \right \}


Option: 2

\left \{ 0,1,2 \right \}


Option: 3

\left \{ 0 \right \}


Option: 4

an empty set 


Answers (1)

best_answer

\\ f o g=f(g(x))=f(1-|x|) \\ =-1+|1-| x|-2| \\ =-1+|-| x|-1|=-1+|| x|+1| \\ \text { Let } f o g=y \\ \therefore y=-1+|| x|+1|

\\\Rightarrow y=\left\{\begin{array}{ll} -1+x+1, & x \geq 0 \\ -1-x+1, & x<0 \end{array}\right.\\ \Rightarrow y=\left\{\begin{array}{ll} x, & x \geq 0 \\ -x, & x<0 \end{array}\right.

\\\text{LHL at }(x=0)=\lim _{x \rightarrow 0}(-x)=0\\ \text{RHL at }(x=0)=\lim _{x \rightarrow 0}(x)=0\\ \text{When } x=0, \text{ then y}=0\\\text{ Hence, } \mathrm{LHL \;at } (x=0)=\mathrm{RHL\;at}\; \;(x=0)=\text{ value of y at } (x=0)\\ \text{Hence y is continuous at }x=0.

Clearly at all other points y continuous. Therefore, the set of all points where fog is discontinuous is an empty set.

Posted by

Kuldeep Maurya

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