Edexcel A Level Further Maths: Core Pure:复习笔记4.1.2 Logarithmic Forms of Inverse Hyperbolic Functions

What are the definitions of the inverse hyperbolic functions?

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  • Since coshx is a many-to-one function, its domain is restricted to x ≥ 0 when finding the inverse
  • Therefore, its range is coshx ≥ 1
  • So, the domain of the inverse function is x ≥ 1
• 
• These three definitions are in the formula booklet

What are the graphs of the inverse hyperbolic functions and their key features?

• As they are inverse functions, they are reflections of their original functions in the line y=x
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  • Domain:
  • Range:

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  • Domain: 
  • Range: 

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  • Domain:
  • Range:

How do I derive the logarithmic formulae for the inverse hyperbolic functions?

• You need to be able to derive each inverse from the definition of the original
• STEP 1
  Write in terms of e
  • and rearrange to
  • 
• STEP 2
  Form a quadratic in terms of ey
  • Multiply by 2ey and rearrange
  • 
• STEP 3
  Solve the quadratic and find an expression for y
  • 
    • Reject  as this produces negative values as  whereas
  • 
• The derivations of the other two formulae are similar
  • For both  and  are well-defined for
    • We choose  as  and it can be shown that