Let for all real .

MATHEMATICS · Multi-part · AP · AP Precalculus · Trigonometric and Polar Functions · Trigonometric and Polar Functions

Problem

Let

f(x)=\arctan(2x)

for all real

x

. (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)

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

g

. (iii) Find all points where the graphs of

f

and

g

intersect. (iv) Sketch the graphs of

f

and

g

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

g

, state the intervals on which

g

is increasing and compare this with the monotonicity of

f