2025 ACJC P1 Q8

The curve with equation \(y = \mathrm{f}(x)\) for \(x \le b\), where \(\mathrm{f}(x)\) is a quadratic function, intersects the \(x\)-axis at the points \((a, 0)\) and \((b, 0)\), and the \(y\)-axis at the point \((0, c)\). The graph of \(y = \mathrm{f}(x)\) is shown in Figure \(1\). The scales on the \(x\)- and \(y\)-axes are the same.

1. Graph Sketching
   1. Sketch the graph of \(y = \mathrm{f}(|x|)\), labelling the points where the graph intersects or touches the axes.[2]
   2. Sketch the graph of \(y = \frac{1}{\mathrm{f}(x)}\), labelling the points where the graph intersects or touches the axes, as well as the equations of any asymptotes.[2]
2. The function \(y = \mathrm{g}(x)\) is such that \(\mathrm{g}(x) = \mathrm{f}(x)\) for \(x \le k\), and \(y = \mathrm{g}^{-1}(x)\) exists.
   1. Find the largest possible value of \(k\) in terms of \(a\) and \(b\).
   2. Using the value of \(k\) found in (b)(i), sketch the graph of \(y = \mathrm{g}^{-1}(x)\) on Figure \(1\) in the Answer Booklet, labelling the intersections with the axes. Write down the range of \(\mathrm{g}^{-1}\) in terms of \(a\) and \(b\).
   3. Explain why the solution to \(\mathrm{g}^{-1}(x) = x\) satisfies the equation \(\mathrm{g}^{-1}(x) = \mathrm{g}(x)\).