The functions of $f(x) = -\tan^{-1}\left(\frac{1}{2}x\right)$ and $g(x) = \cos(x + 90^\circ)$ for $-180^\circ \leq x \leq 180^\circ$ are given - NSC Mathematics - Question 6 - 2016 - Paper 2

To create the sketch of the functions:

1. Identify Characteristics of Each Function:

   • For $f(x) = -\tan^{-1}\left(\frac{1}{2}x\right)$:
     • As $x \to -\infty$, $f(x) \to -\frac{\pi}{2}$
     • As $x \to \infty$, $f(x) \to \frac{\pi}{2}$
     • The function is sigmoid shaped.
   • For $g(x) = \cos(x + 90^\circ) = -\sin(x)$:
     • The amplitude is 1 and the period is $360^\circ$.
     • It oscillates between -1 and 1.

2. Calculate Intercepts:

   • For $f(x)$:
     • The function crosses the y-axis at $f(0) = 0$ (intercept).
   • For $g(x)$:
     • The function crosses the y-axis at $g(0) = 0$ (intercept).

3. Plot the Graphs:

   • Draw both functions on the same axes using a grid.
   • Clearly indicate the intercept points and turning points:
     • The turning points of $g(x)$ occur at multiples of $90^\circ$.
     • Mark the intercept points on the axes clearly.

4. Label Everything Clearly:

   • Title your graph, label axes, and indicate turning points and intercepts clearly.