Modulus Graphs and Inequalities Q1

Functions \(\mathrm{f}\) and \(\mathrm{g}\) are defined as:

\({\mathrm{f}(x) = 4a^2 - x^2,\quad x\in\mathbb{R},}\)
\({\mathrm{g}(x) = \left\vert{}4x - a\right\vert{},\quad x\in\mathbb{R},}\)

where \(a\) is a constant such that \(a\ge 1\).

1. Sketch on the same diagram the graph of \(\mathrm{f}\left( x \right)\) and the graph of \(\mathrm{g}\left( x \right)\). The sketch must include the coordinates of any points where each graph meets the coordinate axes and the exact \(x\) coordinates of intersections in terms of \(a\).
2. Find, in exact form where appropriate, the solutions of the equation:
   \(4-{{x}^{2}}=\left| 4x-1 \right|\).