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)$x = \frac{3}{2} \pm \sqrt{\frac{25}{4} - y}$, $\mathrm{D}_{g^{-1}} = \mathrm{R}_g = \left(-\infty, \frac{25}{4}\right]$ (c)does not exist

2025 JPJC P1 Q4

2025 JPJC P1 Q4