Sequences of Transformations

Recall that we learned about 3 kinds of transformation: Reflection, Rotation, translation. Take a minute to review them if you want.

Sometimes we apply more than one kind of transformation to a figure. Compare the following examples:

Imagine a skateboard sliding down the flat pavement, as shown in the doodle below. The skateboard didn't rotate or reflect, its location change is a result of translation. Only translation is involved in this transformation.

Now, let's take a closer look at the wheels of the skateboard.

The movement of the wheels involve two transformation: translation and rotation. The wheels undergo a sequence of transformations.

Try the question below to test your understanding.

A triangle in the coordinate plane has gone through the following sequence of transformation: $\\$

1. Rotate the triangle around its center by $-90^{\circ}$ $\\$
2. Reflect the triangle over the x-axis $\\$
3. Translate the triangle 5 units to the right.

The identity is the transformation that does nothing. It maps every element in the domain to itself. Unsurprisingly, it is often denoted $I:x\mapsto x$.

The composition of two transformations is the transformation equivalent to doing one transformation first, and then the other. Keep in mind that composition is generally order-sensitive.

For example, the composition $f\circ g$ is the composition formed by applying $g$ first and then $f$. Notice that the order is reversed. To remember this, one may remember $f\circ g$ as $f(g(x))$ and notice that the name of the function applied first shows up farthest to the right.

The identity function is not so interesting by itself, but it has many important properties related to composition.

The identity function, unlike most functions, is not order sensitive: $I\circ f=f\circ I=f$.

The inverse of a function, $f^{-1}$, is the function $g$ such that $I=g\circ f$. The inverse function is also not order-sensitive, so $f^{-1}\circ f=f\circ f^{-1}=I$. Note that the domain of $f$ is the range of $f^{-1}$, and that the range of $f$ is the domain of $f^{-1}$.

Here are two other important properties of function composition, both of which are not difficult to prove: $$(f\circ g)\circ h=f\circ (g\circ h) (associativity)$$ $$(f\circ g)^{-1}=g^{-1}\circ f^{-1}$$