# Function Transformations - how do I find the invariant points?

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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\$.

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### Kat

Updated on January 10, 2020

• 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
• 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.