2025 YIJC Promo Q11 [Modified]

The function \(\mathrm{f}\) is defined by

\[\mathrm{f}: x \mapsto \frac{1}{x^2 + 1}, \quad x \in \mathbb{R}, \: x \le 0.\]

1. Sketch the graph of \(\mathrm{f}\) and explain why the function \(\mathrm{f}^{-1}\) exists.[2]
2. Find \(\mathrm{f}^{-1}(x)\) and state the domain of \(\mathrm{f}^{-1}\).[3]

The function \(\mathrm{g}\) is defined by

\[\mathrm{g}: x \mapsto \mathrm{e}^{-x} - 1, \quad x \in \mathbb{R}, \: x \le 1.\]

3. Find the value of \(a\) such that \(\mathrm{g}^{-1}(a) = 2\).[1]
4. Explain why the composite function \(\mathrm{gf}\) exists.[1]
5. Find \(\mathrm{gf}(x)\) and state the domain of \(\mathrm{gf}\).[2]

6. Find the exact range of \(\mathrm{gf}\).

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