Consider the functions \(f(x)= x^2-1\) and \(g(x)= \ln(x)\). Define the composite function \(h(x)= g(f(x)) = \ln(x^2-1)\).
(a) Determine the domain of \(h(x)\).
(b) Assuming \(h(x)\) is invertible on an appropriately restricted domain, find the expression for its inverse function \(h^{-1}(x)\).
(c) Discuss briefly how the domain restrictions of \(h(x)\) affect the existence of an inverse function.
Question
Determine the domain of the composite function \(h(x)= \ln(x^2-1)\). Give your answer in interval notation.
Hint
Hint
Hint
Set the inequality \(x^2-1 > 0\) and solve for \(x\).
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Answer
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- Formula Booklet
Complete Answer
The domain of \(h(x)\) is \(x \in (-\infty, -1) \cup (1, \infty)\).
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Question
Assuming \(h(x)= \ln(x^2-1)\) is invertible on a restricted domain, find the expression for its inverse function \(h^{-1}(x)\) on that domain.
Hint
Hint
Hint
Set \(y = \ln(x^2-1)\) and solve for \(x\) by exponentiating both sides, then choose the appropriate branch.
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Answer
Answer
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Explanation
Explanation
- Formula Booklet
Complete Answer
\(h^{-1}(x) = \sqrt{e^x+1}\)
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Question
Discuss briefly how the domain restrictions of \(h(x)= \ln(x^2-1)\) affect the existence of an inverse function. In your answer, explain any limitations that arise from the composite function’s domain.
Hint
Hint
Hint
Consider the fact that \(h(x)\) is not one-to-one without restricting its domain.
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Answer
Answer
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Mark Scheme
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Explanation
Explanation
- Formula Booklet
Complete Answer
The inverse exists only on a restricted domain where \(h(x)\) is one-to-one, such as \(x>1\) or \(x<-1\).
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