In this optional video, I'll say a little bit about finding fixed points using Algebra.
So, a fixed point, remember, is just a point that doesn't change when the function acts on it.
In other words, it's a function that...it's a number...
a fixed point is a number x that when the function acts on it, returns x. So x is not changed.
It's fixed, it doesn't change. This is called the "fixed point equation".
So, if we have a function, we can solve for fixed points sometimes by using the fixed point equation.
So, lets do an example.
Suppose we have f of x is one half x minus four.
So then I write down the fixed point equation and I solve for x. So the fixed point equation is:
f of x equals x
And I know that f of x is half x minus four.
And I will solve for x.
So I need to get all of the x's on the same side of the equation. And lets see...
Maybe I'll subtract half x from each side.
So half x minus half x - that goes away - I'm left with minus four...
x minus half x, one minus a half is a half...
And then if I multiply this equation through by two, I get that minus eight equals x.
So that should be my solution.
So the fixed point is minus eight equals x.
Lets check and see if that's really the case.
So lets see. Does f of minus eight equal minus eight? Well, what's f of minus eight?
It's a half (times) minus eight, minus four, wanna know: does that equal minus eight? Lets see...
Half of minus eight, that's minus four... Minus four minus four...does that equal minus eight? Yes.
Thus, minus eight is fixed. So it's not hard to check to see if a number is a fixed point.
You just plug it in, let the function act on it and see if you get out the same number you started with.
Here we do, so we conclude that yes, minus eight really is a fixed point.