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Question

If the functions g(x)={x2,1x2x+2,2<x3

and f(x)={x+4,x12x+1,1<x2

then, the number of roots fo the equation f(g(x))=0 is

A
0
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B
1
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C
2
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D
3
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Solution

The correct option is A 0

g(x)={x2,1x2x+2,2<x3

f(x)={x+4,x12x+1,1<x2

f(g(x))={g(x)+4,g(x)12g(x)+1,1<g(x)2

When 1x1,
then g(x)=x20g(x)1

When 1<x2,
then g(x)=x21<g(x)2

f(g(x))={x2+4,1x1,0g(x)12(x2)+1,1<x21<g(x)2

f(g(x))>0 for x[1,2]


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