Inverse function

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  • #1
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Hello
how can we know with the definition of inverse function

d159c87655554a4e60674ef275482f05.png


this function is inverse function or not?

trix}%20x%28t+1%29%20&%20x%28t%29\geqslant%200\\%200%20&%20x%28t%29\leq%200%20\end{matrix}\right.gif
?

another questions.

gif.gif

this function is inverse function because
gif.gif

but if we have this function
gif.gif

we can't do the same approach.why?(and it isn't a inverse function)

question 3:
in this function
gif.gif

why we can't get Integral and say x1=x2?
 

Answers and Replies

  • #2
Simon Bridge
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I always thought an inverse function was one that undoes another function ... eg. if g(f(x))=x the g is the inverse function of f.

The definition above seem to be saying that there must be only one value for f for each value of x ... eg. f is invertable. But it cannot confirm that some other function g is that inverse - it can only refute it.

For your first example ... y is an inverse function of x if y(x(t))=t
To use the condition for invertability - you can attempt to show that x or y is invertable from the information supplied.
x is invertable if x(t=t1) = x(t=t2) means that t1 = t2

But even if they are invertable, this will not show that y is the inverse of x.
It can help to look closely at the exact question to check what you are being asked to do.
 
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  • #3
140
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Thanks smimon for explanation
it is possible and could you solve the problems with your way? how did you know that y(x(t))=t?

sorry im not celever as you
 
  • #4
Simon Bridge
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how did you know that y(x(t))=t?
Definition of the inverse.

If y(t) is the inverse function of x(t) then y(x(t))=t
it is possible and could you solve the problems with your way?
Probably - but you need to be exact about what the problems want.
As they are written I can't tell.
 
  • #5
chiro
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The answer is no if x(t) is defined to be 0 at more than one point since it maps all these points to one point.

As mentioned earlier, a function has inverse it it is 1-1: in other words, every value produced for the input generates an output that is different to every other input.

You can restrict the inverse to a subset of the domain, but this means that you are looking at a different function since the domain is different (even if the nature of the mapping is the same, since functions require information about the exact nature of the domain and codomain).

Also one thing is that if you know the derivatives of x(t) then you can definitely answer the question with more precision since the inverse function theorem says that an inverse doesn't exist around the neighbourhood of a function mapping when the Jacobian is zero (for 1-D function, this is just the first derivative).
 
  • #6
Simon Bridge
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The answer is no if x(t) is defined to be 0 at more than one point since it maps all these points to one point.
Though that could be a typo...
 

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