# Let $$f:\{1,3,4\} \to \{1,2,5\}and\;g:\{1,2,5\}\to\{1,3\}$$ be given by $$f = \{(1, 2), (3, 5), (4, 1)\} \;and\; g = \{(1, 3), (2, 3), (5, 1)\}$$. Write down $$gof$$.

$\begin{array}{1 1} gof =\{(1,4),(3,1),(4,3)\} \\ gof =\{(1,4),(3,1),(4,3)\} \\ gof =\{(1,3),(3,1),(4,5)\} \\gof =\{(1,3),(3,2),(4,3)\} \end{array}$

Toolbox:
• Given two functions $f:A \to B$ and $g:B \to C$, then composition of $f$ and $g$, $gof:A \to C$ by $gof (x)=g(f(x))\;for\; all \;x \in A$
It is given that $f: \{1,3,4\} \to \{1,2,3 \}$ and $g: \{1,2,5\} \to \{1,3 \}$
$\Rightarrow$ The functions $f, g$ are defined by $f= \{(1,2),(3,5),(4,1)\}$ and $g=: \{(1,3),(2,3),(5,1)\}$
We know that given two functions $f:A \to B$ and $g:B \to C$, then composition of $f$ and $g$, $gof:A \to C$ by $gof (x)=g(f(x))\;for\; all \;x \in A$
$\Rightarrow$ $gof(1)=g(f(1))=g(2)=3$
$\Rightarrow$ $gof(3)=g(f(3))=g(3)=1$
$\Rightarrow$ $gof(4)=g(f(4))=g(1)=3$
From the above syeps we see that the composition of two functions gof is defined by $gof =\{(1,3),(3,1),(4,3)\}$
edited Mar 19, 2013