> [!NOTE] Definition (Function composition) > Given two [[Function|functions]] $f:X\to Y$ and $g:Y\to Z,$ their composition is the function $g\circ f:X\to Z$ defined by $(g\circ f)(x)=g(f(x)).$ *Proof that relation giving $(g\circ f)$ is graphical*. Take $x\in X$ there is a unique $y\in Y$ so that $f(x)=y$ since the relation giving $f$ is graphical. Similarly there is a unique $z\in Z$ so that $g(y)=z$ so $(x,g(f(x)))$ is indeed graphical. # Properties By [[Associativity of Function Composition]], $f\circ(g\circ h) =(f\circ g) \circ h.$