Inverse Functions (1 Viewer)

[jaychouf4n]

Cambridge 3U 1A 18c)

The function f(x) is defined by f(x)=x-1/x

By completing the square or using the quadratic formula show that

inverse (f(x))=1/2(x+root(4+x^2)))

 

[jpmeijer]

Consider:



For the inverse, swap x and y:





, which is a quadratic in y.

Using the quadratic formula:



But now we need to decide whether to take the plus or the minus.
Try substituting a point into the original function. eg let x = 1, which gives y = 0
Therefore (1, 0) lies on the original function.

This means that (0, 1) lies on the inverse.
So if we let x = 0 in , then it is clear to get y = 1 we need to take the positive case.

So therefore:

 

[jaychouf4n]

OMG im a retard.... i screwed up the making a square.... quadratic formula is sooo much easier XD

thanks alot I didn't know about the positive thing You've given me a new technique

 

[Trebla]

Both the positive and negative case are valid, you just have to choose one of them, so that you get a function with 1-1 correspondance. The point (-1, 0) or (0, -1) on the inverse gives the negative case. In other words, either positive or negative case is correct unless some restriction was stated in the question.

 

[jpmeijer]

Both the positive and negative case are valid, you just have to choose one of them, so that you get a function with 1-1 correspondance. The point (-1, 0) or (0, -1) on the inverse gives the negative case. In other words, either positive or negative case is correct unless some restriction was stated in the question.

Yeah my bad, thanks Trebla.