Q.Explain how the graph of $y=\cos^{-1}x$ is obtained from the graph of $y=\cos x$ restricted to $[0,\pi]$.

Restricted to $[0,\pi]$, $\cos x$ is a strictly decreasing bijection onto $[-1,1]$. Since $\cos^{-1}$ is its inverse on this branch, every point $(a,\cos a)$ on the restricted cosine graph corresponds to the point $(\cos a,a)$ on the graph of $\cos^{-1}x$ -- exactly the coordinates swapped. Geometrically, swapping the coordinates of every point on a curve is the same as reflecting that curve in the line $y=x$, so the graph of $y=\cos^{-1}x$ is …

Use the general fact that the graph of any inverse function is the reflection of the original (rest …

Reflecting the entire unrestricted cosine graph, over all $x$, rather than only the principal branch $[0,\pi]$ -- that would not give th …