VCE Methods Integral Calculus Application Task 6

Question 1 [2022 Exam 2 Section B Q4]

Consider the function \( f : \left( -\frac{1}{2}, \frac{1}{2} \right) \rightarrow \mathbb{R} \), \( f(x) = \log_e\left(x + \frac{1}{2}\right) - \log_e\left(\frac{1}{2} - x\right) \).

Part of the graph of \( y = f(x) \) is shown below.

a. State the range of \( f(x) \). 1 mark

b. i. Find \( f'(0) \). 2 marks

ii. State the maximal domain over which \( f \) is strictly increasing. 1 mark

c. Show that \( f(x) + f(-x) = 0 \). 1 mark

d. Find the domain and the rule of \( f^{-1} \), the inverse of \( f \). 3 marks

Let \( h \) be the function \( h : \left( -\frac{1}{2}, \frac{1}{2} \right) \rightarrow \mathbb{R} \), \( h(x) = \frac{1}{k} \left( \log_e\left(x + \frac{1}{2}\right) - \log_e\left(\frac{1}{2} - x\right) \right) \), where \( k \in \mathbb{R} \) and \( k > 0 \).

The inverse function of \( h \) is defined by \( h^{-1} : \mathbb{R} \rightarrow \mathbb{R} \), \( h^{-1}(x) = \frac{e^{kx} - 1}{2(e^{kx} + 1)} \).

The area of the regions bound by the functions \( h \) and \( h^{-1} \) can be expressed as a function, \( A(k) \). The graph below shows the relevant area shaded.

e. i. Determine the range of values of \( k \) such that \( A(k) > 0 \). 1 mark