Is it possible to sketch this function without a graphing calculator?

[Jules18]

x/[(1+3x)^(1/2)-1]

I'm wondering if it's even possible to imagine what a graph of this fxn would look like, or do you definitely need a graphing calc?

When I plug it into a TI83, it ends up looking pretty linear and apparently doesn't exist when x < 0

~Jules~



PS. sorry for how messy the eq'n looks, it's difficult to type. ... (1+3x) should be under a sqrt. sign.

 

[Bohrok]

Just by looking at it, you see that you have x in the numerator and x^1/2 in the denominator. For large x, the function will roughly look like x^1/2, but you can find a better approximation. After multiplying the top and bottom of the fraction by the conjugate of the denominator and doing some work on it, you get

$$\frac{\sqrt{3x + 1} + 1}{3} = \frac{\sqrt{3x(1 + \frac{1}{3x})} + 1}{3} = \frac{\sqrt{3x}\sqrt{1 + \frac{1}{3x}}}{3} ~+~ \frac{1}{3} = \frac{\sqrt{3}}{3}\sqrt{x} \sqrt{1 + \frac{1}{3x}} ~+~ \frac{1}{3}$$

As x→∞, the larger radicand goes to 1 and has less and less effect on √x, so the function is nearly like $$\frac{\sqrt{3}}{3}\sqrt{x} ~+~ \frac{1}{3}$$

 

[Jules18]

thanks ^_^