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December 8th, 2016, 06:24 PM   #1
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Inverse function

Hi,

I know how to find the inverse of a function, but I don't understand why the method calls for the switch of x and y at the end:

so for example we have y=3x+4
my understanding is that the inverse function essentially does the opposite, so instead of the y value being dependent on the x values , the x value will be dependent on the y, so our aim is to get the x on the left hand side
y=3x+4
3x=y-4
x=(y-4)/3

I don't really understand why from this point the final answer is f(x)^-1=
(x-4)/3

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December 8th, 2016, 07:05 PM   #2
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$y=3x+4$ ...

if $x=5$, then $y=19$, so the function forms the ordered pair, $(5,19)$

$y=\dfrac{x-4}{3}$ ...

if $x=19$, then $y=5$, so this function forms the ordered pair, $(19,5)$

What happened to $(x,y)$ from the first function to the second?
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December 8th, 2016, 07:40 PM   #3
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Okay I know exactly what you are asking. I wondered this myself too.

Basically, at first you have y defined by x. The inverse function defines x with y. Makes sense, just re-arrange the equation where x is defined by y.

BEFORE: y=3x+4
AFTER: x=(y-4)/3

So I'm sure you understand this much. But the reason why you should switch the variables at this point is because you want x to be the input and y to be the output. You very well could leave the equation be and let y be your input and x be your output. But we are not used to that and it is hard to understand like that, so that is why we usually switch the two variables at the end, so that y can be the output, rather than the input.

Please ask if you have any questions with what I just said!
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December 8th, 2016, 10:43 PM   #4
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Quote:
Originally Posted by z183 View Post
I don't understand why the method calls for the switch of x and y at the end:
Maybe your (version of the) method does, but it's not that way in two universities I've tutored
the students in. We interchange x and y at the beginning for our preference.


Quote:
Originally Posted by VisionaryLen View Post
..., so that is why we > > > usually < < < switch the two variables at the end, so that y can be the output, rather than the input.
You can speak for yourself, but not "usually," unless you have some strong anecdotal evidence.


.
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Last edited by Math Message Board tutor; December 8th, 2016 at 10:51 PM.
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December 10th, 2016, 06:36 AM   #5
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If function f maps x to y: f(x)= y

then its inverse maps y to x:

If, for example, f(x)= 3x+ 4, equivalent to y= 3x+ 4, then 3x= y- 4 so x= (y- 4)/3.
That is, just as f maps x to y= 3x+ 4, maps y to x= (y- 4)/3.

Swapping x and y comes from our notational convention of always writing a function in the form "y= f(x)". That is, we want the function writing in terms of x so that instead of writing "" we write . Those two equations really mean the same thing.
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December 10th, 2016, 08:50 PM   #6
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Quote:
Originally Posted by Country Boy & Math Message Board tutor edit View Post
If function f maps x to y: f(x)= y

then its inverse $\displaystyle f^{-1}$ maps y to x: $\displaystyle f^{-1}(y)= x$

If, for example, f(x)= 3x+ 4, equivalent to y= 3x+ 4, then 3x= y- 4 so x= (y- 4)/3.
That is, just as f maps x to y= 3x+ 4, $\displaystyle f^{-1}$ maps y to x= (y- 4)/3.

Swapping x and y comes from our notational convention of always writing a function in the form "y= f(x)".
That is, we want the function writing in terms of x so that instead of writing "$\displaystyle f^{-1}(y)= (y- 4)/3$" we write
$\displaystyle f^{-1}(x)= (x- 4)/3$. Those two equations really mean the same thing.
Country Boy, in the quote box, I replaced "tex" with "math" and "/tex" with "/math" in your
text so the Latex would work with on this forum.

Please make the necessary adjustments for your Latex posting.

Last edited by Math Message Board tutor; December 10th, 2016 at 08:55 PM.
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