Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by\[\mathrm{f} : x \mapsto [\ln(x - 1)]^2 +

Question

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by$$\mathrm{f} : x \mapsto [\ln(x - 1)]^2 + 2, \quad x \ge a,$$$$\mathrm{g} : x \mapsto 4 + 3x - x^2, \quad x \le \frac{3}{2}.$$ It is given that the function $\mathrm{f}^{-1}$ exists. State the smallest value of $a$.[1] Find an expression for $\mathrm{g}^{-1}(x)$, stating its domain.[3] Using the value of $a$ found in part (a), determine whether the composite function $\mathrm{g}^{-1}\mathrm{f}^{-1}$ exists.[1]

Accepted Answer

(a) $a=2$. (b) $\mathrm g^{-1}(x)=\frac{3}{2}-\sqrt{\frac{25}{4}-x}$, domain $( -\infty,\frac{25}{4}]$. (c) The composite does not exist because $[2,\infty)\not\subseteq(-\infty,\frac{25}{4}]$.

2025 JPJC P1 Q4

2025 JPJC P1 Q4

Solution:

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