The function $\mathrm{f}$ is defined by \[\mathrm{f} : x \mapsto \frac{1}{x^2 + 2x}, \quad x \in \

Question

The function $\mathrm{f}$ is defined by $$\mathrm{f} : x \mapsto \frac{1}{x^2 + 2x}, \quad x \in \mathbb{R}, \quad x < -1, \quad x \neq -2$$ Show that $\mathrm{f}$ has an inverse.[1] Define $\mathrm{f}^{-1}$ in similar form.[4] The function $\mathrm{g}$ is such that $\mathrm{fg}(x) = x^2 + 1$. Find $\mathrm{g}(x)$.[2] The function $\mathrm{h}$ is defined by $$\mathrm{h} : x \mapsto \frac{1}{x^2}, \quad x \in \mathbb{R}, \quad x \neq 0$$ Only one of the composite functions $\mathrm{fh}$ and $\mathrm{hf}$ exists. Give a definition (including the domain) of the composite that exists, and explain why the other composite does not exist.[3]

Accepted Answer

(a) $f^{-1}$ exists. (b) $f^{-1}:x\mapsto-1-\sqrt{1+\frac1x}$, $x0$. (c) $g(x)=-1-\sqrt{1+\frac1{x^2+1}}$. (d) $hf:x\mapsto(x^2+2x)^2$, $x<-1$, $x\ne-2$.

2025 YIJC P1 Q11

2025 YIJC P1 Q11

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