Question

How I find the range of the following, without graphing? \(\ \Huge \mathsf{\frac{x}{3x-1}} ? \)

Answer 1

well the range is everywhere that y is defined.. Since this is a fraction the range is defined where the denominator is not 0.. so set 3x-1= 0 and solve for x

Answer 2

So...

Answer 3

That gives me the \(\ \mathsf{Domain}\) though, which is \(\ \Huge \mathsf{(-infin,\frac{1}{3})U(\frac{1}{3}, infin)}...\)

Answer 4

you need to consider extreme values.
consider this:
\[\frac{x}{3x -1} = \frac{1}{3 - \frac{1}{x}}\]

what happens when x has a very large magnitude?

Answer 5

The 1/x gets closer to infity, so it's ~1/3

Answer 6

does it ever equal 1/3 ?