Composition of Other Algebraic Functions ( especially Linear Fractional Functions)
As we have seen with the linear and quadratic functions, viewing a function as a composition can enhance understanding of the function.
We continue here to explore how composition can be used to understand some of the more frequently encountered rational functions.

For viewing general rational functions dynamically refer to Example OAF.DRF.0 :Dynamic Visualization for a Rational Function.

Two simple examples.
In these examples we use GeoGebra to visualize the composition for one linear core function with the core negative power function, $R(x) = \frac 1 x$.
Notice how poles and asymptotic behavior are effected by the composition.
Example OAF.COMP.1 : $ f(x) =  \frac 1 x; g(x) = x-2$
Example OAF.COMP.2 :$ f(x) = x - 2; g(x) = \frac 1 x$
Two more examples.
In these examples we use GeoGebra  to visualize the composition for some quadratic polynomial functions with the core negative power function, $R(x) = \frac 1 x$.
Notice how poles and asymptotic behavior are effected by the composition.
Example OAF.QCOMP.1 : $ f(x) =  \frac 1 x; g(x) = x^2-1$
Example OAF.QCOMP.2 : $ f(x) =  \frac 1 x; g(x) = x^2+1$
Compositions are a key to understanding many other algebraic functions. The following key result on linear fractional functions as compositions is very useful.
Theorem LFF.COMP

This result says linear fractional functions are built by composing the core negative power function, $R(x) = \frac 1 x$ with core linear functions, $f_{-h}, f_k$, and $f_{*A}$.
Here is an example that visualizes the theorem.

Example OAF.COMP.3 Suppose $f$ is a linear fractional function with $f(0) = -1$, pole at $x=1$ and $f(2)=3$.
Verify that $f =f_{+1}∘f_{∗2}∘R∘f_{−1} $.
Visualize $f$ with a mapping diagram that illustrates the function as the composition: $f = f_{+1}∘f_{∗2}∘R∘f_{−1}$.

Symmetry of  Linear Fractional Functions: Suppose $f(x) =\frac A {x-h} + k$.

Understanding $f$ with compositions and mapping diagrams explains why linear fractional functions always have odd symmetry with respect to the point ($(h,k)$).

You can use this next dynamic example with GeoGebra to investigate further the symmetry of a  linear fractional function in a mapping diagram  of $f$.

Example OAF.DSYMM.0 Dynamic Visualization of Symmetry for  Linear Fractional Functions: Graphs, and Mapping Diagrams