VCE Methods Integral Calculus Application Task 20

Question 1 [2017 Exam 2 Section B Q4]

Let \(f: \mathbb{R} \to \mathbb{R} : f(x) = 2^{x+1} - 2\). Part of the graph of \(f\) is shown below.

a. No longer in the curriculum

b. Find the rule and domain for \(f^{-1}\), the inverse function of \(f\). 2 marks

c. Find the area bounded by the graphs of \(f\) and \(f^{-1}\). 3 marks

d. Part of the graphs of \(f\) and \(f^{-1}\) are shown below.

Find the gradient of \(f\) and the gradient of \(f^{-1}\) at \(x = 0\). 2 marks

The functions of \(g_k\), where \(k \in \mathbb{R}^+\), are defined with domain \(\mathbb{R}\) such that \(g_k(x) = 2e^{kx} - 2\).

e. Find the value of \(k\) such that \(g_1(x) = f(x)\). 1 mark

f. Find the rule for the inverse functions \(g_k^{-1}\) of \(g_k\), where \(k \in \mathbb{R}^+\). 1 mark

g.

i. Describe the transformation that maps the graph of \(g_1\) onto the graph of \(g_k\). 1 mark

ii. Describe the transformation that maps the graph of \(g_1^{-1}\) onto the graph of \(g_k^{-1}\). 1 mark

h. The lines \(L_1\) and \(L_2\) are the tangents at the origin to the graphs of \(g_k\) and \(g_k^{-1}\) respectively.
Find the value(s) of \(k\) for which the angle between \(L_1\) and \(L_2\) is 30°. 2 marks

i. Let \(p\) be the value of \(k\) for which \(g_k(x) = g_k^{-1}(x)\) has only one solution.

i. Find \(p\). 2 marks

ii. Let \(A(k)\) be the area bounded by the graphs of \(g_k\) and \(g_k^{-1}\) for all \(k > p\).
State the smallest value of \(b\) such that \(A(k) < b\). 1 mark