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

Question

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.

Answer 1

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 $(5,2)$.

Answer 2

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?