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Question

# Let f(x)=1x,g(x)=14x2−1 and h(x)=5xx+2 be three functions and k(x)=h(g(f(x))). If domain and range of k(x) are R−{a1,a2,a3,…,an} and R−A respectively, where R is the set of real numbers, then

A
n+ni=1ai=5
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B
n+ni=1ai=10
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C
Number of integers in set A is 5
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D
Number of integers in set A is 7
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Solution

## The correct options are A n+n∑i=1ai=5 D Number of integers in set A is 7k(x)=h(g(f(x)))=5x28−x2 Domain of k(x) is R−{0,±2,±2√2} Let y=5x28−x2 Then, (y+5)x2−8y=0 As x is real, so D≥0 and y+5≠0 ⇒8y(y+5)≥0 and y≠−5 ⇒y∈(−∞,−5)∪[0,∞) But y=0 for x=0 and y=5 for x=±2 which are not in the domain of k(x). ∴ Range of k(x) is (−∞,−5)∪(0,∞)−{5} or R−([−5,0]∪{5}) ∴A=[−5,0]∪{5}

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