I have this question, as below, and I cannot understand what I am being asked of. Typically I would consider the composition of functions to be rather simple, such as $f(x)=2x-5$ $g(x)=x-2$, therefore $f(g(x))=2x-9$. However, this question does not resemble that of what I am familiar with.
Let $X=\{1,2,3,4\}$. The functions $f$ and $g$ (from the set $X$ to the set $X$) are $f=\{[1,2],[2,2],[3,4],[4,3]\}$ and $g=\{[1,3],[2,1],[3,4],[4,1]\}$. Determine the compositions $f∘g$ and $g∘f$. Give your answers in the same form as $f$ and $g$ above. The composed function $f∘g$ is the function that at the point $x$ gets the value $f(g(x))$. The notation above for the functions $f$ and $g$ means that, for example, $f(2)=2$.
Can someone please explain what '$f=\{[1,2],[2,2],[3,4],[4,3]\}$' means and then how to begin finding the composition between these two functions.
A function can be given as a set of ordered pairs. The function you write as "$f(x)=2x-5$", if we don't mind restricting its domain to a small set, say $\{0,1,2\}$, can be expressed as $f=\{[0,-5],[1,-3],[2,-1]\}$. This is just a compact way of giving us the values of $f$ for each domain point.
In your example, you want to write $f\circ g$ in the same way. Thus, you want to write:
$f\circ g=\{[1,f(g(1))],[2,f(g(2))],[3,f(g(3))],[4,f(g(4))]\}$
Each of those second elements in the pairs must be calculated, so for example, $f(g(1))=f(3)=4$. We know that $g(1)=3$, because $g$ includes the pair $[1,3]$, and likewise for $f(3)$. Consequently, the first pair in the set defining $f\circ g$ should be $[1,4]$.
Does that clarify what's being asked here?