Find inverse for f(x)= x/(1+x)

[jimjames]

Homework Statement

Show that the functions f are one-to-one and calculate the inverse function.

Homework Equations

f(x)= x/(1+x) (It is the equation I am having trouble with)

The Attempt at a Solution

I know the solution is y= x/(1-x) But no idea how to solve it.

My attempt:
x(1+y)=y or x+xy=y

y+y=x/x

 

[Mark44]

Homework Statement

Show that the functions f are one-to-one and calculate the inverse function.

Homework Equations

f(x)= x/(1+x) (It is the equation I am having trouble with)

The Attempt at a Solution

I know the solution is y= x/(1-x) But no idea how to solve it.

My attempt:
x(1+y)=y or x+xy=y

Start with $y = \frac x {1 + x}$
What's the first thing you need to do?

y+y=x/x

I'm not sure what you did here.

 

[jimjames]

Where did you get y = x/(1+x) ?

What i normally do is
f(x) = x/(1+x) <=> x=f^-1(y)
f(y) = y/(1+y) <=> y=f^-1(x)

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

And now I'm stuck

 

[Mark44]

Where did you get y = x/(1+x) ?

What i normally do is
f(x) = x/(1+x) <=> x=f^-1(y)
f(y) = y/(1+y) <=> y=f^-1(x)

The above doesn't help you with this problem. All you're doing is manipulating symbols.

x=y/(1+y)

In the line above, all you did was switch x and y.

y=x+xy

How did you get the equation above?

And now I'm stuck

Start with y = $\frac x {1 + x}$

For this problem DO NOT switch the variables x and y.
DO solve for x in the equation just above. In other words, x should appear only on one side of the equation.

 

[jimjames]

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

 

[SammyS]

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

OK.

Now divide both sides by x.

 

[Mark44]

Where did you get y = x/(1+x) ?

That's the function you're trying to find the inverse of.
Multiplying both sides of this equation by 1 + x gives you the equation just below.

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

OK.

Now divide both sides by x.

It's probably simpler to expand y(1 + x), get all terms that involve x on one side, and then isolate x.

 

[Ray Vickson]

Homework Statement

Show that the functions f are one-to-one and calculate the inverse function.

Homework Equations

f(x)= x/(1+x) (It is the equation I am having trouble with)

The Attempt at a Solution

I know the solution is y= x/(1-x) But no idea how to solve it.

My attempt:
x(1+y)=y or x+xy=y

y+y=x/x

Why do you call this a "transcendental function"? It is just about as far from transcendental as you can get.

 

[Mark44]

Why do you call this a "transcendental function"? It is just about as far from transcendental as you can get.

I changed the title a while ago for that very reason.

 

[HallsofIvy]

Where did you get y = x/(1+x) ?

What i normally do is
f(x) = x/(1+x) <=> x=f^-1(y)
f(y) = y/(1+y) <=> y=f^-1(x)

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

And now I'm stuck

Subtract xy from both sides: y- xy= y(1-x)= x

 

[jimjames]

Managed to solve this late yesterday.
Thanks for trying to help.