Problem
ALG-B3-M08-P002 Image of an Idempotent
#2
★★☆☆☆ Level 2 of 5
Let \(f(f(x))=f(x)\) for all \(x\). Prove that every element in the image of \(f\) is a fixed point.
Write an image element as \(f(t)\).
Let \(y\) be in the image. Then \(y=f(t)\) for some \(t\). Therefore \(f(y)=f(f(t))=f(t)=y\). Hence \(y\) is fixed.
Teaches how to read \(f^2=f\).