On the same axes, sketch the graphs of $y=\frac{x-b}{x}$ and $y=\frac{1}{a}\left| x-b \right|$,

Question

On the same axes, sketch the graphs of $y=\frac{x-b}{x}$ and $y=\frac{1}{a}\left| x-b \right|$, where $0<a<1$ and $b>1$. State the equations of any asymptotes and the coordinates of the points where the curves cross the axes.[3] Hence solve the inequality $\frac{x-b}{x}\le \frac{1}{a}\left| x-b \right|$.[3]

Accepted Answer

(i) For $y=\frac{x-b}{x}$: asymptotes $x=0$, $y=1$, intercept $(b,0)$. For $y=\frac1a|x-b|$: intercepts $(b,0)$, $(0,\frac ba)$. (ii) $x\le-a$ or $x>0$.

2022 NJC Promo Q3

2022 NJC Promo Q3

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