Functions $\mathrm{f}$ and $\mathrm{g}$ are respectively defined on the domain of real numbers by $\

Question

Functions $\mathrm{f}$ and $\mathrm{g}$ are respectively defined on the domain of real numbers by $\mathrm{f}:x\mapsto {{x}^{2}}-2x+2$, $x>1$ $\mathrm{g}:x\mapsto \sqrt{x+3}$ , $x\ge -3$ By considering the derivative of $\mathrm{f}(x)$, prove that $\mathrm{f}$ is a one-one function. [2] Solve $\mathrm{f}(x)={{\mathrm{f}}^{-1}}(x)$ where ${{\mathrm{f}}^{-1}}(x)$ denotes the inverse functions of $\mathrm{f}(x)$. [2] Deduce the solution set to the inequality $\mathrm{f}(x)\le {{\mathrm{f}}^{-1}}(x)$. [2] Show that the composite function $\mathrm{gf}$ exists and define it in a similar form. State also its range. [3]

Accepted Answer

(i) $\mathrm{f}$ is strictly increasing function for $x>1$, hence, $\mathrm{f}$ is a one-one function. (ii) $x=2$ (iv) ${{R}_{\mathrm{gf}}}=\left( 2,\infty \right)$

2010 NYJC P1 Q7

2010 NYJC P1 Q7

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