Inverse fn q (1 Viewer)

Axio · Oct 25, 2015

Can inverse functions intersect at points not on the line y=x?

InteGrand · Oct 25, 2015

Axio said:
Can inverse functions intersect at points not on the line y=x?

$Yes. If the graph of $y=f(x)$ happens to pass through the points $A(a,b)$ and $B(b,a)$ with $|a|\neq |b|$, then the graph of $y=f^{-1}(x)$ will also pass through the points $A$ and $B$, and these aren't on the lines $y=\pm x$.$

InteGrand · Oct 25, 2015

InteGrand said:
$Yes. If the graph of $y=f(x)$ happens to pass through the points $A(a,b)$ and $B(b,a)$ with $|a|\neq |b|$, then the graph of $y=f^{-1}(x)$ will also pass through the points $A$ and $B$, and these aren't on the lines $y=\pm x$.$

$It can be proved though that any such invertible function $f$ that is continuous on $[a,b]$ (where $a<b$) will cross the line $y=x$, and thus will intersect the graph of $y=f^{-1}(x)$ on the line $y = x$ as well. So if a continuous invertible function intersects its inverse, at least one of the intersection points must be on the line $y=x$. But all of them need not be.$

Physicklad · Oct 25, 2015

note that either f(x) or f-(x) cannot be a function if the two intersect at a point that is not on the line y=x

i.e. if both function and inverse are functions the only points of intersection lie on y=x

InteGrand · Oct 25, 2015

Physicklad said:
note that either f(x) or f-(x) cannot be a function if the two intersect at a point that is not on the line y=x

i.e. if both function and inverse are functions the only points of intersection lie on y=x

$This isn't true, for example consider the function $f(x) = -x$. The inverse is $f^{-1} (x) = -x$, which intersects the graph of $y=f(x)$ in infinitely many places that are not on the line $y=x$.$

godofindolence · Oct 25, 2015

InteGrand said:
$Yes. If the graph of $y=f(x)$ happens to pass through the points $A(a,b)$ and $B(b,a)$ with $|a|\neq |b|$, then the graph of $y=f^{-1}(x)$ will also pass through the points $A$ and $B$, and these aren't on the lines $y=\pm x$.$

So does this mean that the inverse function is exactly same as the actual function?

InteGrand · Oct 25, 2015

godofindolence said:
So does this mean that the inverse function is exactly same as the actual function?

$Not necessarily, e.g the function $f(x) = -x^3$ has inverse $f^{-1} (x) = -\sqrt[3]{x}$, and these intersect at two points that are not on the line $y=x$ (the points $(-1,1)$ and $(1,-1)$), but are not the same function.$

laters · Oct 25, 2015

Some functions have their inverses as themself e.g. f(x)=1/x... they would 'intersect' at all real non-zero x on the curve itself.

Inverse fn q (1 Viewer)

Axio

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