Question

# Let g(x) = {−xx≤1x+1x≥1 and f(x) = {1−xx≤0x2x≥0 Consider the composition of f and g. i.e, (f ∘ g)(x) -f(g(x)). The number of discontinties is (f ∘ g)(x) present in the interval (-∞,0) is.

A
0
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B
1
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C
2
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D
4
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Solution

## The correct option is A 0g(x) = {−xx≤1x+1x≥1 f(x) = {1−xx≤0x2x≥0 f ∘ g(x) = f(g(x)) = {1−g(x),g(x)≤0(g(x))2,g(x)≥0 =⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩1−(−x)−x≤0,x≤11−(x+1),x+1≤0,x≥1(−x)2−x>0,x≤1(x+1)2,x+1>0,x≥1 = ⎧⎪⎨⎪⎩1+x,0≤x≤1x2x<0(x+1)2x>1 f(g(x)) = ⎧⎪⎨⎪⎩x2x<0x+1,0≤x≤1(x+1)2x>1 f ∘ g(x) is continuous at every point in (-∞, 0). ⇒ Number of points of discontinuities = 0 Overall we have two points of discontinuities at x = 0 and x = 1 but these do not belong to the given interval (-∞, 0).

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