For x ≠0,1, define f1(x)=x , f2(x)=(1)/(x), f3(x)=1-x f4(x)=(1)/(1-x), f5(x)=(x-1)/(x)

step 1 So in this video we're going to be discussing composition of functions. So there's a lot of ways

For x ≠0,1, define f1(x)=x , f2(x)=(1)/(x), ...

For $x \neq 0,1,$ define $$\begin{aligned}&f_{1}(x)=x , \quad f_{2}(x)=\frac{1}{x}, \quad f_{3}(x)=1-x\\&f_{4}(x)=\frac{1}{1-x}, f_{5}(x)=\frac{x-1}{x}, f_{6}(x)=\frac{x}{x-1}\end{aligned}$$ This family of functions is closed under composition; that is, the composition of any two of these functions is again one of these functions. Tabulate the results of composing these functions one with the other by filling in the table shown in the figure. To indicate that $f_{i} \circ f_{j}=f_{k},$ write "$f_{k}$" in the $i$th row, $j$th column. We have already made two entries in the table. Check out these two entries and then fill in the rest of the table.

For $x \neq 0,1,$ define
$$\begin{aligned}&f_{1}(x)=x , \quad f_{2}(x)=\frac{1}{x}, \quad f_{3}(x)=1-x\\&f_{4}(x)=\frac{1}{1-x}, f_{5}(x)=\frac{x-1}{x}, f_{6}(x)=\frac{x}{x-1}\end{aligned}$$
This family of functions is closed under composition; that is, the composition of any two of these functions is again one of these functions. Tabulate the results of composing these functions one with the other by filling in the table shown in the figure. To indicate that $f_{i} \circ f_{j}=f_{k},$ write "$f_{k}$" in the $i$th row, $j$th column. We have already made two entries in the table. Check out these two entries and then fill in the rest of the table.

Key Concepts

Fractional Linear (Möbius) Transformations

The functions given are examples of fractional linear transformations (or Möbius transformations), which are functions of the form (ax + b)/(cx + d) with ad - bc ? 0. These transformations are significant in many areas of mathematics, including complex analysis and geometry, because they preserve the cross-ratio of points and exhibit rich symmetry properties. Their behavior under composition often leads to nicely structured algebraic systems.

Functional Composition

Functional composition is the process of applying one function to the result of another function, denoted (f ? g)(x) = f(g(x)). This concept is fundamental in many areas of mathematics as it describes the building of complex functions from simpler ones, and the particular problem leverages this idea by composing provided functions to see how they combine.

Closure Under Composition and Group Structure

A set of functions is said to be closed under composition if the composition of any two functions in the set is also a function within the set. This closure is one of the key properties of a mathematical group. In the context of the problem, the six functions form a set that is closed under composition, indicating that they might possess a group structure with an identity element, inverses, and associativity.

Cayley Table

A Cayley table is a tabular representation of the operation (here, function composition) within a set. Each cell in the table represents the result of composing two functions. In problems like this one, the Cayley table not only organizes the outcomes but also can reveal algebraic properties such as commutativity and the existence of inverses, thereby providing insight into the structure of the group formed by the functions.

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