Finding Inverse Functions?

[neowmoko]

I'm trying to work through this problem, and it's giving me a lot of trouble:
Given g(x) = (√x) / (√x) + 1, find the inverse function g^-1(x).
Heres where I am at so far:
I start by setting the equation to equal y like so:
y = (√x) / (√x) + 1
Then I need to solve for x, so I multiply the bottom half by y so I get this:
y((√x) + 1) = √x
After that I multiply the left side out and get to this:
y√x + y = √x
I divide √x out of the left hand side getting:
2y = √x/√x
2y = 1
y = 0.5
I don't feel like I got the right answer by doing this, and was wondering if anyone knows where I went wrong?
Or maybe I got it correct? It just doesn't feel like y would equal 0.5

 

[Dr.Peterson]

I'm trying to work through this problem, and it's giving me a lot of trouble:
Given g(x) = (√x) / (√x) + 1, find the inverse function g^-1(x).
Heres where I am at so far:
I start by setting the equation to equal y like so:
y = (√x) / (√x) + 1
Then I need to solve for x, so I multiply the bottom half by y so I get this:
y((√x) + 1) = √x
After that I multiply the left side out and get to this:
y√x + y = √x
I divide √x out of the left hand side getting:
2y = √x/√x
2y = 1
y = 0.5
I don't feel like I got the right answer by doing this, and was wondering if anyone knows where I went wrong?
Or maybe I got it correct? It just doesn't feel like y would equal 0.5

Thanks for showing work, just as we ask!

First, it appears that you don't really mean g(x) = (√x) / (√x) + 1, which would mean g(x) = 2 (for non-negative x). Rather, you appear to mean g(x) = √(x) / (√(x) + 1). Do you see the difference?

Second, when you said this,

y√x + y = √x​

I divide √x out of the left hand side getting:​

2y = √x/√x​

you weren't thinking carefully. You just divided the first term by √x, not the entire left side. That is illegal.

Try again.

 

[Deleted member 4993]

I'm trying to work through this problem, and it's giving me a lot of trouble:
Given g(x) = (√x) / (√x) + 1, find the inverse function g^-1(x).
Heres where I am at so far:
I start by setting the equation to equal y like so:
y = (√x) / (√x) + 1
Then I need to solve for x, so I multiply the bottom half by y so I get this:
y((√x) + 1) = √x
After that I multiply the left side out and get to this:
y√x + y = √x
I divide √x out of the left hand side getting:
2y = √x/√x
2y = 1
y = 0.5
I don't feel like I got the right answer by doing this, and was wondering if anyone knows where I went wrong?
Or maybe I got it correct? It just doesn't feel like y would equal 0.5

y = √x/[√x+1].........................those parentheses are super important.

1/y = 1 + 1/√x

[y/(1-y)]² = x ......................continue

 

[Steven G]

You clearly said that you need to solve for x, but then solved for y.

You need to understand that if y is given then no matter what you do y will not change. How can you go from y = (√x) / (√x) + 1 to y = 0.5?!!

 

[pka]

Lets suppose that
$y=\dfrac{\sqrt x}{\sqrt x-1}\\y\sqrt x-y=\sqrt x\\-y=\sqrt x-y\sqrt x\\y=y\sqrt x-\sqrt x\\\dfrac{y}{y-1}=\sqrt x$
Can you finish this off?