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#1 2006-11-22 17:47:50

Toast
Real Member
Registered: 2006-10-08
Posts: 1,321

What's an inverse function?

This term seems to be used a lot in the website, but I can't say i've heard it mentioned in class.
Please explain and give an example.

Last edited by Toast (2006-11-22 17:49:35)

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#2 2006-11-22 18:57:43

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: What's an inverse function?

For a function, there may exist an f-¹ such that:

f(f-¹(x)) = x and f-¹(f(x)) = x

Note, that this is not guaranteed to exist.

Some examples where it does exist:

f(x) = 2x and f-¹(x) = (1/2)x
f(x) = sqrt(x) and f-¹(x) = x^2, for all x greater than 0

Some where it doesn't:

f(x) = x^2, for all x in R
f(x) = sin(x), for all x


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#3 2006-11-22 19:25:19

luca-deltodesco
Member
Registered: 2006-05-05
Posts: 1,470

Re: What's an inverse function?

carrying on from what ricky said, there is always an inverse function, however, the inverse function is not defined over the entire range that the function is defined over. For example.

f(x) = sin(x), f-¹(x) = sin-¹(x), for -1/2pi < x < 1/2pi, and outside of that range, the inverse does not hold.

there exist a continuous inverse function, if the function is also continiuois, and is one to one, i.e. for every value the function can take, there is only one value of x that can give it, otherwise the inverse has to be restricted to a domain.


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#4 2006-11-22 22:53:13

Toast
Real Member
Registered: 2006-10-08
Posts: 1,321

Re: What's an inverse function?

thanks, but nevermind lol I don't really understand what you're talking about

Last edited by Toast (2006-11-22 22:54:13)

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#5 2006-11-23 03:44:06

luca-deltodesco
Member
Registered: 2006-05-05
Posts: 1,470

Re: What's an inverse function?

well to find an inverse function you do this:

for example:

f(x) = 2x^2+2

let y = f(x)

y = 2x^2 + 2, now you have to rearrange it for x

2x^2 = y-2
x^2 = 0.5y - 1
x = sqrt(0.5y - 1)

then you switch y for x

f-¹(x) = sqrt(0.5x - 1)

now looking at the original function, f(x) is one-one for x<=0 or x>=0, and there the domains you can use when taking the inverse


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#6 2006-11-23 06:34:27

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: What's an inverse function?

carrying on from what ricky said, there is always an inverse function, however, the inverse function is not defined over the entire range that the function is defined over.

This is correct.  However, very (let me emphasize very) strictly speaking, two functions are different if their domains differ.  That is:

f(x) = x^2, for all x in R

As opposed to:

g(x) = x^2, for all x in R, x > 0

Are different functions with different properties.  f, for example, is reflexive over the y axis.  g obviously is not.

An inverse, I believe, is defined over the entire domain that the function it is, and thus, the inverse may not be a function.

But really, at this point, we're just splitting hairs for the most part.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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