The functions of $f(x) = -\tan^{-1}\left(\frac{x}{2}\right)$ 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 sketch the graphs of the functions, we need to analyze them individually.

1. Function $f(x) = -\tan^{-1}\left(\frac{x}{2}\right)$:

   • The range of $f(x)$ is between $-\frac{\pi}{2}$ and $0$.
   • x-intercept occurs when $f(x) = 0$: $\tan^{-1}\left(\frac{x}{2}\right) = 0 \Rightarrow x = 0.$
   • The function is continuous and monotonically increasing in the interval.

2. Function $g(x) = \cos(x + 90^\circ)$:

   • This simplifies to $g(x) = -\sin(x)$.
   • The y-intercept occurs when $x=0$: $g(0) = -\sin(90^\circ) = -1.$
   • The x-intercepts can be calculated as:
     • For $g(x) = 0 \Rightarrow -\sin(x) = 0 \Rightarrow x = n\pi$ for integers $n$.

3. Plotting: On the same set of axes, plot $f(x)$ and $g(x)$, marking the x and y intercepts, as well as any turning points derived from their derivatives. Ensure to label each graph clearly and use appropriate scales for the axes.