The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}:x\mapsto {{x}^{2}}+4x-5\mathr

Question

The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}:x\mapsto {{x}^{2}}+4x-5\mathrm{ }$ for $x\in \mathbb{R},\,\,x\le k$, $\mathrm{g}:x\mapsto \ln \left( x+10 \right)\mathrm{ }$ for $x\in \mathbb{R},\mathrm{ }x>-9$. Determine the largest value of $k$ for which f is a one-one function. With this value of $k$, define the inverse function of $\mathrm{f}$ in similar form. [4] Given that $k=-3$ show that the composite function gf exists and state its range, [3] solve the inequality $\left| \mathrm{f}(x) \right|+2x+3\ge 0$, leaving your answer in exact form. [4]

Accepted Answer

(a) $k=-2$, $f^{-1}(x)=-2-\sqrt{x+9}$ for $x\ge-9$. (b)(i) $R_{gf}=[\ln2,\infty)$. (b)(ii) $x\le-3-\sqrt{11}$ or $-4\le x\le-3$.

2013 RVHS P2 Q3

2013 RVHS P2 Q3

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