It is given that ${\mathrm{f}(x) = \frac{x^2 - 2}{x}}$, verify that \({\left\vert{}\mathrm{f}(x)\r

Question

It is given that ${\mathrm{f}(x) = \frac{x^2 - 2}{x}}$, verify that ${\left\vert{}\mathrm{f}(x)\right\vert{} = \left\vert{}\mathrm{f}(\left\vert{}x\right\vert{})\right\vert{}}$. Hence, use this equation to describe the graph ${y = \left\vert{}\mathrm{f}(x)\right\vert{}}$. [2] Sketch the curve with equation $y=\left\vert{}\frac{x^2-2}{x}\right\vert{}$, giving the exact coordinates of the point(s) where the curve crosses the axes and the equations of any asymptotes.[4] Solve the inequality $\left\vert{}\frac{x^2-2}{x}\right\vert{} < x+4$ exactly.[3]

Accepted Answer

(i) $|f(x)|=|f(|x|)|$; the graph is symmetric about the $y$-axis. (ii) $(\pm\sqrt2,0)$; no $y$-intercept; asymptotes $x=0$, $y=-x$ as $x\to-\infty$ and $y=x$ as $x\to\infty$. (iii) $-1-\sqrt2\sqrt2-1$.

2022 TGM P1 Q8

2022 TGM P1 Q8

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