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How to sketch $y = \frac1{\sqrt{x-1}}$

My way:(which does not work here)

I normally solve these problems by squaring and converting them to equations of 2 degree curves(such as parabola, hyperbola, etc.) which I can easily plot. But this seems to go 3 degree as $xy^2$ term is coming.

Please help me to solve this.

Note: Please don't say to use a graph plotter and see for myself since in the exam if this question comes I won't have the graph plotter with me.

user2369284
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  • Sketch the graph of $y=\sqrt x$. Shift it to the right $1$ unit (to get the graph of $y=\sqrt{x-1}$). Invert this graph to obtain the graph of your function. (Transformation techniques are sometimes very handy.) – David Mitra Dec 24 '13 at 17:31

2 Answers2

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Consider some basic properties of the function, which you can work out either by inspection or by considering derivatives:

  • It is only defined for $x \ge 1$;
  • It has no roots, stationary points, inflection points, etc.;
  • It is always decreasing and convex;
  • It tends to $0$ as $x \to \infty$;
  • It tends to $\infty$ as $x \to 1^+$.

Just this information is enough for you to give a rough sketch of the function.

If you want to make it more accurate then you could consider some points which the function passes through, e.g. $(2,1)$ and $(5,2)$.

Clive Newstead
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  • @CliveNewsteed How did you decide that this curve is always decreasing and convex ? – user2369284 Dec 24 '13 at 16:04
  • @user2369284: By looking at the sign of $y'$ and $y''$ (or just by inspection: there's nothing else it could do!). – Clive Newstead Dec 24 '13 at 16:05
  • @CliveNewsteed And how does the sign of $y^{'}$ and $y^{''}$ matter. – user2369284 Dec 24 '13 at 16:07
  • Because $y'$ tells you whether the function is increasing or decreasing (if $y'>0$ then the function is increasing and if $y'<0$ then it's decreasing), and $y''$ tells you whether the function is convex or concave (likewise). – Clive Newstead Dec 24 '13 at 16:08
  • @CliveNewsteed Can't we analyse the curve for $y = x^{-0.5}$ and then shift the origin to (1,0). – user2369284 Dec 24 '13 at 16:10
  • @user2369284: Yes, you can do that... but how was I meant to know that you knew how to sketch that function? The method I suggested would work even if you didn't know how to sketch that one. – Clive Newstead Dec 24 '13 at 16:11
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HINT : What is the domain of $x$? What happens if you make $x$ larger, larger, to infinity? What happens if you make $x$ closer to $1$ from the right side?

mathlove
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