**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.

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.

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.

Compositions are a key to understanding many other algebraic functions. The following key result on linear fractional functions as compositions is very useful.

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: $f = f_{+1}∘f_{∗2}∘R∘f_{−1}$.

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 $f$.