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Question

What is the Domain and Range of inverse hyperbolic tangent function?

A
Domain (0,1) and Range [0,)
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B
Domain (1,1) and Range (,)
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C
Domain (,1) and Range (,)
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D
Domain (1,) and Range [1,)
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Solution

The correct option is B Domain (1,1) and Range (,)
tanh1x=12ln1+x1x
1x>0
x<1
also, 1+x1x>0
x+1x1<0
x(1,1)
For range,
y=12ln1+x1x
2y=ln1+x1x
e2y=1+x1x
(1x)e2y=1+x
e2yxe2y=1+x
e2y1=x(1+e2y)
e2y1e2y+1=x
It is satisfied for all values of y. Hence y(,)

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