<s>[INST] Range of a rational function with radicals Find the range of the function 
$$\frac{6}{5\sqrt{x^2-10x+29} - 2}$$
I tried using inverses, but the equation got super messy and I dont think its a good method for this problem.
$\frac{6}{5\sqrt{x^2-10x+29} - 2} = y$ 
getting the inverse,
$\frac{6}{5\sqrt{y^2-10y+29} - 2} = x$ 
$\frac{4x^2+24x+36}{25x^2}= y^2-10y +29$
Then it would be a quadratic function in y,  but the discriminant becomes really big
$100- 116(\frac{4x^2+24x+36}{25x^2})$
 [/INST] $\frac{6}{5\sqrt(x^2-10x+29)-2}=\frac{6}{5\sqrt((x-5)^2+4)-2}$
so you can see that there will be a maximum at $x=5$ and as $x$ tends to either positive or negative infinity the graph goes to zero.
So just plug in $x=5$ to get $\frac{3}{4}$ and you have a range $]0,\frac{3}{4}]$
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