2016 VCE Maths Methods Mini Test 8

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.

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