# Prove a function is onto

• Apr 12th 2009, 01:51 PM
SlapnutsGT
[Solved]Prove a function is onto
I need to prove $f(x) = ((x+1)/x)$ is an onto function. I'm kind of hazy here, am I supposed to subsitute and solve for x? Such as, $y=((x+1)/x)$. IF that is the case, I can't get it to add up to prove that function is onto. I know answer to problem, its just I think I'm jackng up my algebra and want to be able to do all the steps involved.

Thanks!
• Apr 12th 2009, 01:55 PM
Plato
Onto what?
What is the domain?
What is the image set?
• Apr 12th 2009, 02:00 PM
SlapnutsGT
Doh! Sorry all real numbers, domain is x not equal to 0 and co-domain not equal to 1.
• Apr 12th 2009, 02:17 PM
spearfish
Here's what you do:
Suppose f is onto. Then there exists an x such that f(x) = (x+1)/(x) = y. Solve for x to get x = 1/(y-1). Now plug this value back into the equation, like this:
f(1/(y-1)) = (x+1)/(x) = [(1/(y-1))+1]/[1/(y-1)]. Now solve this equation to end up with f(1/(y-1)) = y. There, now you have proved that f(x) is onto.
• Apr 12th 2009, 02:31 PM
SlapnutsGT
Quote:

Originally Posted by spearfish
Here's what you do:
Suppose f is onto. Then there exists an x such that f(x) = (x+1)/(x) = y. Solve for x to get x = 1/(y-1). Now plug this value back into the equation, like this:
f(1/(y-1)) = (x+1)/(x) = [(1/(y-1))+1]/[1/(y-1)]. Now solve this equation to end up with f(1/(y-1)) = y. There, now you have proved that f(x) is onto.

That is what I was orignally doing. I know the problem is I'm messing up my algebra. I can't solve for x, I always end up cancelling my x-variables. Sorry, my algebra is rusty.

y=((x+1)/x) ...multiply both sides by x
xy=x+1 ...subtract 1
xy-1=x ...divide by (y-1)
x=x/(y-1)

Stuck here, not even sure if I did those steps properly.
• Apr 12th 2009, 02:37 PM
spearfish
Yeah, I hear you, it happens to me too!

y = (x+1)/x
y = x/x + 1/x
y = 1 + 1/x
-1/x = 1-y
1/x = -1+y
1 = x(-1+y)
1/(y-1) = x

Hope that helps.
• Apr 12th 2009, 02:44 PM
SlapnutsGT
Quote:

Originally Posted by spearfish
Yeah, I hear you, it happens to me too!

y = (x+1)/x
y = x/x + 1/x
y = 1 + 1/x
-1/x = 1-y
1/x = -1+y
1 = x(-1+y)
1/(y-1) = x

Hope that helps.

Thats is what I was forgetting right there, thanks alot!