How to plot $\frac{\sqrt{12+x-x^2}}{x(x-2)}$

Question

How to plot $\frac{\sqrt{12+x-x^2}}{x(x-2)}$

My solution:

The roots are : -3,4

Domain : x = [-3,4] - {0,2}

The derivative is coming very large so is there any other way to do it?

How to proceed further ? Please help.

Answer 1

Besides to the points which @mathlover posted, you may take a limit of the function as follows:

$$\lim_{x\to\pm\infty}f(x)=0$$ so the functions face the ground when $x$ tends to infinity. Also regarding to the roots of the denominator we get $$x\to 0^+\longrightarrow x(x-2)\to 0^-\\x\to 0^-\longrightarrow x(x-2)\to 0^+\\x\to 2^+\longrightarrow x(x-2)\to 0^+\\x\to 2^-\longrightarrow x(x-2)\to 0^-$$ Eventually, we can collect all points we got to predict the plot. I think we can escape of differentiation here. Without doing that, we can feel how the plot behaves.

Answer 2

The derivative is large, but we need to do that in order to plot the function.

Let $f(x)$ be the function. Also, let $g(x)=-x^2+x+12.$

Then, we have

$$f^\prime (x)=\frac{\frac{(-2x+1)x(x-2)}{2\sqrt{g(x)}}-\sqrt{g(x)}(2x-2)}{x^2(x-2)^2}=\frac{x(x-2)(-2x+1)-2(2x-2)g(x)}{x^2(x-2)^2\cdot 2\sqrt{g(x)}}$$

So, you should know how to treat the numerator.