C3 inverse function question

hi for part c of this question, why can the inverse function be expressed as x when it is root x+3 ?


Thanks

hi for part c of this question, why can the inverse function be expressed as x when it is root x+3 ?


Thanks

Because the inverse is the reflection in y=x of the function (the dotted line on the diagram). Hence, the place that f^(-1)(x) and f(x) intersect is also the place that they intersect with the line f(x)=x.

This is the point of them saying that "this is easier than solving ..." because solving f(x) = x or f^(-1)(X)=x will return the same point.

Hope that's clear.

How do you know if they intersect or not at y=x ? Thanks

That's a thing. The inverse of any function will be the reflection in y=x. Hence they have to intersect there. Its just like if you touch a mirror, your inverse (reflection) touches the mirror at the same place.


Here is a good discussion
http://math.stackexchange.com/questions/387542/is-there-an-explanation-why-the-reflection-of-fx-through-y-x-is-its-invers

How do you know if they intersect or not at y=x ? Thanks

How do you know if they intersect or not at y=x ? Thanks

A snappier but less geometric demonstration is thus:

Suppose $P(a,b)$ is the intersection of $f, f^{-1}$.

Then so is the point $P'(b,a)$, since $f : a \mapsto b \Rightarrow f^{-1}: b \mapsto a$ by definition of the inverse. So $P=P' \Rightarrow (a,b)=(b,a) \Rightarrow a=b$.

So we have $P(a,a)=P'(a,a)$ is the point of intersection, and therefore $P,P'$ lie on $y=x$