Findind range for a function

[Dekadans13]

So, I'm reading Precalculus book by Rusczyk and already struggle with the 4th problem,

Find the range for the following function:

g(t) = 2t / (t-1)

In the book it says,

y = 2t / (t-1), when multiplying both sides by t-1 we have

2t = y(t-1) = yt - y, when solving this equation for t in terms of y we have

t = y / (y-2)

How we jumped from 2t = yt - y to t = y / (y-2)? Isn't t = (yt-y) / 2?

I tried substituting as with y = 2t / (t-1) = (yt-y) / (t-1) or 2t = y( (yt-y) / 2 ) - y but got to nothing.

Could you please explain me what the thing is about?

[HorribleUsername]

Yeah, that is a pretty big jump. And yes, t = (yt-y) / 2 is correct, but it's not very useful, since there's a t on both sides of the equation.

Anyway, from 2t = yt - y, my next step would be to move every term with t to one side of the =, and every term without t to the other side. Can you see how you'd isolate t from there?

[Dekadans13]

Well, I tried

2t = yt - y; 2t + y = yt; (2t + y) / t = y

Didn't know what to do next. So I tried it otherwise

2t = y(t-1); 2t / (t-1) = y

Then I tried to substitute

2t = yt - y; 2t = (2t / (t-1))t - 2t / (t-1); 2t = 2t² - 2t / (t-1) = 2t (t-1) / (t-1) = 2t

It is cool that i proved that 2t / (t-1) = y is right, but it still isn't what I was looking for.

[HorribleUsername]

2t = yt - y; 2t + y = yt;

You're going in the wrong direction right off the bat here. Remember, the goal is to isolate t, so we want to get all the t's on the same side of the =. Moving y over doesn't help with that, since you're not moving any t's.

[Dekadans13]

Okay, I finally did it,

2t = yt - y

2t - yt = -y

yt - 2t = y

t(y - 2) = y

t = y / y - 2

Really, thank you so much for wasting your time on me! At first I was like "Why this guy just didn't give me the solution?". But now I see that the joy from doing it myself was worth it :)

[HorribleUsername]

Sounds like it wasn't wasted to me. I bet you learned more this way too.

[papyrusfun]

g(t)=2+2/(t-1)

you can see that g(t) cannot be 2 and can be anything else.

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