SOLUTION: For {{{f(x) = 2/(x-3)}}} , solve for the domain and range in interval notation I found already the domain: (-infinity,3) u (3,+infinity) and i'm confident that it is correct

Algebra ->  Functions -> SOLUTION: For {{{f(x) = 2/(x-3)}}} , solve for the domain and range in interval notation I found already the domain: (-infinity,3) u (3,+infinity) and i'm confident that it is correct      Log On


   

Question 1041914: For f%28x%29+=+2%2F%28x-3%29 , solve for the domain and range in interval notation
I found already the domain:
(-infinity,3) u (3,+infinity)
and i'm confident that it is correct.
What confuses me is finding the Range.
I'm not confident on my answer:
(- infinity , 2] u [2, + infinity)
for the + infinity there, i mean it to be positive but approaching zero making 2 as the maximum output that can be obtained in this function.
I'm lost in that part. so can you give me the correct range in INTERVAL NOTATION and explain it efficiently? thanks.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Let's find the inverse of the function.


f%28x%29+=+2%2F%28x-3%29


y+=+2%2F%28x-3%29 Replace f(x) with y


x+=+2%2F%28y-3%29 Swap x and y. Now solve for y.


x%28y-3%29+=+2 Multiply both sides by y-3


xy-3x+=+2 Distribute


xy-3x%2B3x+=+2%2B3x Add 3x to both sides


xy+=+2%2B3x


%28xy%29%2Fx+=+%282%2B3x%29%2Fx Divide both sides by x


y+=+%282%2B3x%29%2Fx


so g%28x%29+=+%282%2B3x%29%2Fx is the inverse of f%28x%29+=+2%2F%28x-3%29

As a side project, I recommend you confirm that f%28g%28x%29%29+=+x and g%28f%28x%29%29+=+x. I'll leave this for you to do on your own.


--------------------------------------------


Why is the inverse so important? Because the domain of the inverse function is the same as the range of the original function. Recall that the domain and range swap when it comes to inverses and their original counter-part functions. The swap occurs because x and y are swapped.


The domain of g(x) is any number but x = 0 to avoid division by zero errors.


So the range of f(x) is the same thing, just with a different variable: y will be any number but y = 0.


The range of f(x) in interval notation is (-infinty, 0) U (0, infinity)