Function Transformations - how do I find the invariant points?

functions

39,822

You're looking for points where $f(x)=x$. Phrased differently, you're looking for points where $(x,x)$ lies on your graph. Rephrased once more, you're looking for intersection with the line $y=x$.

39,822

Kat

Updated on January 10, 2020

Comments

Kat over 3 years

When they give you a picture of a graph (doesn't matter what kind - linear, parabola, inverse, etc.) how do you find the invariant points?
- epimorphic almost 8 years
  
  Possibly related: Analysis Question? Fixed Point? and Fixed point location for functions
- Rock over 5 years
  
  I believe the question is how to determine invariant points between a function and it's inverse. If you're looking to algebraically find the point, you just make the two functions equal each other, and then solve for x.
Kat almost 8 years

I get what I am looking for but how do I find it?
Zach Stone almost 8 years

Draw the line $y=x$, and look for intersection with the given graph.
Kat almost 8 years

Is there any way to do it algebraically?
Zach Stone almost 8 years

Wait, are you given a picture or equation? If you have the equation, set $f(x)=x$ and try to solve for $x$. I can't be more specific without more details about what $f$ looks like. Sometimes it's possible, sometimes it's not.
Kat almost 8 years

It is just a picture of a graph of two porabolas side by side (one is a reflection over the y-intercept) that will eventually intersect but do not in the picture.
Zach Stone almost 8 years

If you don't have any equations, then you can't do any algebra.