Finding the inverse of a rational function

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Homework Statement



Find the inverse of a rational function: [itex]f(x)=\frac{2x}{x-2}[/itex]

The Attempt at a Solution



1. Verify the function is one-to-one.

Question: is there a way to do that withought drawing the graph? Some algebraic method? I know that this means that the function crosess any given x=a line only once, so, e.g. the function with two y intercepts would not be 1-1 ???

2. show that [itex]f(x)=f^{-1}(x)[/itex]

Q.: How to express the given equation in the form where x is given in terms of y? It is imposible since you always get either y or x on both sides.
 

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  • #2
SammyS
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Homework Statement



Find the inverse of a rational function: [itex]f(x)=\frac{2x}{x-2}[/itex]

The Attempt at a Solution



1. Verify the function is one-to-one.

Question: is there a way to do that withought drawing the graph? Some algebraic method? I know that this means that the function crosess any given x=a line only once, so, e.g. the function with two y intercepts would not be 1-1 ???

2. show that [itex]f(x)=f^{-1}(x)[/itex]

Q.: How to express the given equation in the form where x is given in terms of y? It is impossible since you always get either y or x on both sides.
What is [itex]f^{-1}(f(x))\ ?[/itex]
 
  • #3
CAF123
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Q.: How to express the given equation in the form where x is given in terms of y? It is imposible since you always get either y or x on both sides.

You should be able to solve for [itex] x [/itex] in terms of [itex] y. [/itex] You will need to take out a common factor at some stage.
 
  • #4
HallsofIvy
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Homework Statement



Find the inverse of a rational function: [itex]f(x)=\frac{2x}{x-2}[/itex]

The Attempt at a Solution



1. Verify the function is one-to-one.

Question: is there a way to do that withought drawing the graph? Some algebraic method? I know that this means that the function crosess any given x=a line only once, so, e.g. the function with two y intercepts would not be 1-1 ???
Technically, if there are two y intercepts, it would not be a function. That is different from saying it is not one to one. A function is not one to one if it crosses some y line more than once.

To determine whether this function is one to one, look at what happens if two values of x give the same y: suppose 2a/(a- 2)= 2b/(b- 2). Multply on both sided by (a- 2)(b- 2) to get 2a(b- 2)= 2b(a- 2). That is the same as 2ab- 2a= 2ba- 2b. We can subtract 2ba (= 2ab) from both sides to get -2a= -2b, and finally divide both sides by -2: a= b, showing that two different values of x cannot give the same y.

2. show that [itex]f(x)=f^{-1}(x)[/itex]

Q.: How to express the given equation in the form where x is given in terms of y? It is imposible since you always get either y or x on both sides.
That kind of equation is the whole point of algebra! If you have, for example, 4xy= y- x, you solve for x by adding x to both sides: xy+ x= y, factoring out x: x(y+ 1)= 4, and dividing both sides by y+ 1: x= 4/(y+1).
 
  • #5
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You should be able to solve for [itex] x [/itex] in terms of [itex] y. [/itex] You will need to take out a common factor at some stage.

I do it thus:

(i) [itex]y=\frac{2x}{x-2}[/itex]

(ii) [itex]y(x-2)=2x[/itex]

(iii) [itex]\frac{y(x-2)}{2}=x[/itex]

Is this the right path? Probably not ...
 
  • #6
HallsofIvy
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From (ii) y(x- 2)= 2x, multiply out the left side: xy- 2y= 2x, then get the "x"s together on one side- xy- 2x= 2y; x(y- 2)= 2y. Can you finish that?
 
  • #7
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From (ii) y(x- 2)= 2x, multiply out the left side: xy- 2y= 2x, then get the "x"s together on one side- xy- 2x= 2y; x(y- 2)= 2y. Can you finish that?

Thanks, to you and to all :)
 
  • #8
SammyS
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Notice that for the function given,
[itex]f(f(x))=x\ .[/itex]​
 

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