Inverse Functions

[Jason76]

Looking at \(\displaystyle y = mx + b\), inverse \(\displaystyle f^{-1}(y)\) is finding the domain by plugging in the range. So what would be \(\displaystyle f^{-1}(x)\) Is it finding the range by plugging in the domain? In that case it would be the same thing as \(\displaystyle f(x) = mx + b\) same as \(\displaystyle y = mx + b\) Of course, any function including parabolas could have been used instead of the linear function I gave.

You can find \(\displaystyle f^{-1}(y)\) by solving the equation for x rather than y.

\(\displaystyle y = mx + b\)


\(\displaystyle y - b = mx + b - b\)

\(\displaystyle \dfrac{y - b}{m} = \dfrac{mx}{m}\)

\(\displaystyle \dfrac{y - b}{m} = x\)

same as:

\(\displaystyle \dfrac{y - b}{m} = f^{-1}(y)\)

 

[lookagain]

Of course, any function including > > parabolas < < could have been used instead of the linear function I gave.[/tex]

Jason76,

I am choosing to address only that portion of your post.

If you are stating that (e.g. vertical or horizontal) parabola functions have inverses, that is not true.
They are not one-to-one. They fail both the horizontal and vertical line tests, for example.

But, if you were to restrict the domains of the vertical parabolas (or the range of the horizontal
parabolas) sufficiently (such as semi-parabolas), then they would have inverses.


For example, y = x^2 has no inverse. (It fails the horizontal line test.)


But, y = x^2, x >= 0 does have an inverse.

 

[Jason76]

\(\displaystyle f(x) = \) y value. However, there is not \(\displaystyle f(y)\) to find the x value. But there is \(\displaystyle f^{-1}(y)\) to find the x value (domain). Is there a \(\displaystyle f^{-1}(x)\) ? I was thinking that would be the same as \(\displaystyle f(x)\) which gives you the y value (range).

Yes, as the poster stated above, not all parabolas have an inverse.

 

[HallsofIvy]

\(\displaystyle f(x) = \) y value. However, there is not \(\displaystyle f(y)\) to find the x value. But there is \(\displaystyle f^{-1}(y)\) to find the x value (domain). Is there a \(\displaystyle f^{-1}(x)\) ? I was thinking that would be the same as \(\displaystyle f(x)\) which gives you the y value (range).

I really have no idea what you are trying to say here. x, or y, or u, or whatever, is a "placeholder"- a letter representing whatever number you put in there. If \(\displaystyle f^{-1}(y)= \frac{y- b}{m}\), then it is true that \(\displaystyle f^{-1}(x)= \frac{x- b}{m}\), \(\displaystyle f^{-1}(u)= \frac{u- b}{m}\).

Yes, as the poster stated above, not all parabolas have an inverse.

Actually, NO parabolas, that represent functions, have an inverse function.

(It doesn't really make sense to say that a "parabola", a geometric object has, or does not have, an "inverse". It is functions that have, or do not have, inverses.)