Edexcel IAL - Further Pure Mathematics 3- 1.2 Inverse Hyperbolic Functions and Logarithmic Forms- Study notes - New syllabus
Edexcel IAL – Further Pure Mathematics 3- 1.2 Inverse Hyperbolic Functions and Logarithmic Forms -Study notes- New syllabus
Edexcel IAL – Further Pure Mathematics 3- 1.2 Inverse Hyperbolic Functions and Logarithmic Forms -Study notes -Edexcel A level Maths- per latest Syllabus.
Key Concepts:
- 1.2 Inverse Hyperbolic Functions and Logarithmic Forms
Inverse Hyperbolic Functions and Their Graphs
The inverse hyperbolic functions are the inverse functions of \( \sinh x \), \( \cosh x \), and \( \tanh x \). Their graphs are obtained by reflecting the graphs of the corresponding hyperbolic functions in the line \( y = x \), after restricting the original functions to suitable domains.
Main Inverse Hyperbolic Functions
| Inverse Function | Original Function | Restricted Domain of Original Function |
|---|---|---|
| \( \operatorname{arsinh} x \) | \( \sinh x \) | All real values of \( x \) |
| \( \operatorname{arcosh} x \) | \( \cosh x \) | \( x \geq 0 \) |
| \( \operatorname{artanh} x \) | \( \tanh x \) | All real values of \( x \) |
Graphical Features of Inverse Hyperbolic Functions
| Function | Key Graph Features | Graph |
|---|---|---|
| \( y = \operatorname{arsinh} x \) | Odd function; passes through the origin; increasing for all \( x \) |  |
| \( y = \operatorname{arcosh} x \) | Defined only for \( x \geq 1 \); starts at \( (1,0) \); increasing |  |
| \( y = \operatorname{artanh} x \) | Vertical asymptotes at \( x = \pm 1 \); increasing on \( (-1,1) \) |  |
Key Examination Notes
- Graphs are reflections of the corresponding hyperbolic functions in \( y = x \)
- Always state domain restrictions clearly
- Asymptotes and intercepts are commonly tested
Example :
Sketch the graph of \( y = \operatorname{arsinh} x \), stating its key features.
▶️ Answer/Explanation
The function \( y = \operatorname{arsinh} x \) is the inverse of \( y = \sinh x \).
Key features:
Domain: all real numbers
Range: all real numbers
Odd function
Passes through the origin
The graph is obtained by reflecting \( y = \sinh x \) in the line \( y = x \).
Conclusion: The curve increases for all \( x \) and passes smoothly through the origin.
Example :
Sketch the graph of \( y = \operatorname{arcosh} x \) and state its domain.
▶️ Answer/Explanation
The function \( y = \operatorname{arcosh} x \) is the inverse of \( y = \cosh x \), restricted to \( x \geq 0 \).
Key features:
Domain: \( x \geq 1 \)
Range: \( y \geq 0 \)
Passes through \( (1,0) \)
Increasing function
The graph is the reflection of the right-hand branch of \( y = \cosh x \) in the line \( y = x \).
Conclusion: The curve starts at \( (1,0) \) and rises slowly.
Example :
Sketch the graph of \( y = \operatorname{artanh} x \), showing any asymptotes.
▶️ Answer/Explanation
The function \( y = \operatorname{artanh} x \) is the inverse of \( y = \tanh x \).
Key features:
Domain: \( -1 < x < 1 \)
Range: all real numbers
Vertical asymptotes at \( x = \pm 1 \)
Passes through the origin
The graph is obtained by reflecting \( y = \tanh x \) in the line \( y = x \).
Conclusion: The curve increases rapidly near the asymptotes \( x = \pm 1 \).
Inverse Hyperbolic Functions – Properties and Logarithmic Equivalents
Inverse hyperbolic functions are most effectively handled using their logarithmic equivalents. Their key algebraic and graphical properties are frequently tested in examination questions.
Logarithmic Equivalents of Inverse Hyperbolic Functions
| Inverse Hyperbolic Function | Logarithmic Equivalent | Condition |
|---|---|---|
| \( \operatorname{arsinh} x \) | \( \ln \left( x + \sqrt{1 + x^2} \right) \) | All real \( x \) |
| \( \operatorname{arcosh} x \) | \( \ln \left( x + \sqrt{x^2 – 1} \right) \) | \( x \geq 1 \) |
| \( \operatorname{artanh} x \) | \( \dfrac{1}{2}\ln \left( \dfrac{1 + x}{1 – x} \right) \) | \( -1 < x < 1 \) |
Key Properties of Inverse Hyperbolic Functions
| Function | Domain | Range | Key Properties |
|---|---|---|---|
| \( \operatorname{arsinh} x \) | All real numbers | All real numbers | Odd function; passes through the origin; increasing for all \( x \) |
| \( \operatorname{arcosh} x \) | \( x \geq 1 \) | \( y \geq 0 \) | Minimum value \( 0 \) at \( x = 1 \); increasing function |
| \( \operatorname{artanh} x \) | \( -1 < x < 1 \) | All real numbers | Odd function; vertical asymptotes at \( x = \pm 1 \) |
Important Examination Notes
- Logarithmic forms are essential for solving equations and differentiation
- Always check domain restrictions before simplifying
- Inverse hyperbolic graphs are reflections of the original graphs in \( y = x \)
Example :
Show that \( \operatorname{arsinh}(-x) = -\operatorname{arsinh} x \).
▶️ Answer/Explanation
Using the logarithmic equivalent:
\( \operatorname{arsinh}(-x) = \ln\!\left( -x + \sqrt{1 + x^2} \right) \)
Factor out a negative sign inside the logarithm:
\( = \ln\!\left( \sqrt{1 + x^2} – x \right) \)
Since
\( \sqrt{1 + x^2} – x = \dfrac{1}{x + \sqrt{1 + x^2}} \)
we obtain
\( \operatorname{arsinh}(-x) = -\ln\!\left( x + \sqrt{1 + x^2} \right) \)
Conclusion: \( \operatorname{arsinh}(-x) = -\operatorname{arsinh} x \), so \( \operatorname{arsinh} x \) is an odd function.
Example :
Evaluate \( \operatorname{arcosh} 5 \) exactly.
▶️ Answer/Explanation
Using the logarithmic form:
\( \operatorname{arcosh} x = \ln\!\left( x + \sqrt{x^2 – 1} \right) \)
\( \operatorname{arcosh} 5 = \ln\!\left( 5 + \sqrt{25 – 1} \right) \)
\( = \ln(5 + \sqrt{24}) = \ln(5 + 2\sqrt{6}) \)
Conclusion: \( \operatorname{arcosh} 5 = \ln(5 + 2\sqrt{6}) \).
Example :
Solve the equation \( \operatorname{artanh} x = 1 \).
▶️ Answer/Explanation
Using the logarithmic equivalent:
\( \operatorname{artanh} x = \dfrac{1}{2}\ln\!\left( \dfrac{1 + x}{1 – x} \right) \)
So
\( \dfrac{1}{2}\ln\!\left( \dfrac{1 + x}{1 – x} \right) = 1 \)
\( \ln\!\left( \dfrac{1 + x}{1 – x} \right) = 2 \)
\( \dfrac{1 + x}{1 – x} = e^2 \)
\( 1 + x = e^2(1 – x) \)
\( x(1 + e^2) = e^2 – 1 \)
\( x = \dfrac{e^2 – 1}{e^2 + 1} \)
Conclusion: The solution is \( x = \dfrac{e^2 – 1}{e^2 + 1} \), which satisfies \( -1 < x < 1 \).