Inverse of function

1. Feb 19, 2009

Nyasha

1. The problem statement, all variables and given/known data

Find the inverse of y=3x-(1/2x)

3. The attempt at a solution

y=(6x^2-1)/(2x)

x=(6y^2-1)/(2y)

2yx=(6y^2-1)

2yx-6y^2=-1

2y(x-3y)=-1

2y(x-3y)=-1

( I am stuck here how do l solve for "y")

2. Feb 19, 2009

Dick

You've got 6y^2-2xy-1=0. That's a quadratic equation in y. Solve it the way you would any other quadratic equation ay^2+by+c=0. a=6, b=-2x c=-1.

3. Feb 19, 2009

Staff: Mentor

I realize that this switching of x and y is how many (most?) books present this process,
but it's really not necessary and can lead to some confusion on the part of students. The only reason for doing this switching of variables is that both the original function and its inverse can be written as y = f(x) and y = f-1(x). In other words, with y as the dependent variable and x as the independent variable.

That convenience gets in the way, IMO, in much of what you do with inverses later on. For a function that has an inverse, the most important relationship between the two functions is this:
$y = f(x) \Leftrightarrow x = f^{-1}(y)$

This says that if one function gives you a y value associated with an x input value, the other, the inverse, gives you the x value if you know the y value. Because there is no switching of variables, the graph of both functions is exactly the same -- there is no reflection across the line y = x.

A very commonly used example is:
$y = e^x \Leftrightarrow x = ln(y)$

I'm probably fighting a losing battle on this...

4. Feb 20, 2009

Nyasha

Thanks for the help l figured out a way of solving the inverse of the function without using quadratic formula.

5. Feb 20, 2009

HallsofIvy

Staff Emeritus

6. Feb 20, 2009

Nyasha

y=3x-(1/2x)

Multiply entire equation by 2x to get 2xy = 3x – 1.

Add 1 both sides 2xy + 1 = 3x.

Subtract 2xy from both sides, giving 1 = 3x - 2xy

Factor out x giving 1=x(3-2y)

Divided both sides by 3-2y giving x=1/(3-2y)

Interchange x and y giving y= 1/(3-2x)

7. Feb 20, 2009

Dick

2x*(3x-1/(2x))=6x^2-1, not 3x-1. Or was your problem really y=(3x-1)/(2x)?

8. Feb 20, 2009

Nyasha

Ummmmm, thanks for point out that error. This means one way or the other l will have to use quadratic formula

9. Feb 20, 2009

Nyasha

After l solve the quadratic equation l get:

x= (y±sqrt(y²+6))/6

To get the inverse do l now need to interchange "x" and " y" ?

10. Feb 20, 2009

Dick

Sure. Or you could have interchanged them first and solved for y. Either way.

11. Feb 20, 2009

Unco

Are you given some restriction on x (i.e., the domain of f(x) = 3x - 1/(2x)) ? Otherwise, as you have discovered, f(x) does not have an inverse (an inverse function must be single-valued!).

12. Feb 20, 2009

Nyasha

The domain of the original function is : x is an element of real # except zero and y is an element of real #. To get the inverse l am supposed to change the domain and the range, which l have done.

13. Feb 26, 2009

Nyasha

x= (y±sqrt(y²+6))/6

So if my restricted domain is D={x|x<0} do l choose the negative function ?