2022 VCE Maths Methods Mini Test 8

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_eleft(x + frac{1}{2}right) - log_eleft(frac{1}{2} - xright) ).

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_eleft(x + frac{1}{2}right) - log_eleft(frac{1}{2} - xright) 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