Mathematics

Problem

Let

f(x)=\arctan(2x)

for all real

x

. a. (i) The graph of

f

is reflected in the

x

-axis and then translated vertically upward by

\dfrac{\pi}{4}

. Write the equation of the resulting function

g(x)

. b. (ii) Determine the equations of all horizontal asymptotes of the graph of

g

. c. (iii) Find all points where the graphs of

f

and

g

intersect. d. (iv) Sketch the graphs of

f

and

g

on the same set of axes, clearly labelling intercepts and asymptotes. e. (v) From the algebraic form of

g

, state the intervals on which

g

is increasing and compare this with the monotonicity of

f