The function $\mathrm{h}$ is given by \[\mathrm{h}:x\mapsto\mathrm{e}^{\sqrt{x^2-1}},\quad x\in\ma

Question

The function $\mathrm{h}$ is given by $$\mathrm{h}:x\mapsto\mathrm{e}^{\sqrt{x^2-1}},\quad x\in\mathbb{R},\ \lvert x\rvert>a.$$ State the range of values of $a$ for $\mathrm{h}$ to be defined.[1] For the rest of the question, let $a=2$. Show that $\mathrm{h}(x)$ does not have a stationary value.[2] Sketch the graph of $\mathrm{h}$, stating the equation of the line of symmetry.[3] Hence, find the domain over which $\mathrm{h}$ is an increasing function whose range is $(10,\infty)$.[1]

Accepted Answer

(i) $a\ge1$ (ii) $\mathrm{h}'(x)=\dfrac{x\,\mathrm{e}^{\sqrt{x^2-1}}}{\sqrt{x^2-1}}\ne0$ for $\lvert x\rvert>2$ (iii) Symmetric about $x=0$; end points $\left(\pm2,\mathrm{e}^{\sqrt{3}}\right)$ excluded (iv) $\left(\sqrt{1+(\ln10)^2},\,\infty\right)$

2026 NJC Promo Q6

2026 NJC Promo Q6

Solution:

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