Chapter 2.8 Involution
In this section, we will describe the a class of transformations called involutions. Three of the four are transformations we have seen. The last one, called \(y\)-axis inversion, is one transformations that affects the “magnitude” or “size” of a point.
Reflections and Rotation by Half of a Circle
To begin, we define an involution.
Involution
An involution \(\iota\colon \mathbb R^2\to \mathbb R^2\) is a function with the property that \(\iota \circ \iota\) is the identity function \({\rm Id}\).
This is to say that for any \((a,b)\) in \(\mathbb R^2\), \[\iota(\iota(a,b)) = (a,b).\]
Read \(\iota\) as “iota” or “yota”.
In other words, it is a function that when applied twice undos the initial transformation. We have seen some before.
Example 1
Rotations by half a circle are involutions.
Focus primarily on the rotation by half a circle about \((0,0)\) and denote this rotation by \(R\) so that if \((a,b)\) is a point in \(\mathbb R^2\), then \[R(a,b) = (-a, -b).\]
Here are two other transformations we have seen before.
Example 2
For any line \(L\) in the plane, the reflections across \(L\) is an involution.
We will focus primarily on reflections across the \(y\)-axis and reflections across \(x\)-axis.
Denote by \(M_y\) the reflection about the \(y\)-axis and denote by \(M_x\) the rotation across the \(x\)-axis, so that if \((a,b)\) is a point in \(\mathbb R^2\), then \[M_y(a,b) = (-a, b) \quad{\rm and}\quad M_x(a,b) = (a, -b).\]
Practice applying these kind of involutions with a linear function.
Example 3
Take \(f\) to be the function that is given by \[f(x) = x - 5.\] Find equations for the functions given by \(R(f)\), \(M_y(f)\), and \(M_x(f)\) and sketch \(f\) along with these transformed functions.
Recall that \(f=\{(x,f(x))\colon x\in \mathcal{D}(f)\}\).
So \[\begin{align}Rf&=\{R(x,f(x))\colon x\in \mathcal{D}(f)\}\\ &=\{(-x,-f(x))\colon x\in \mathcal{D}(f)\}\\ &=\{(X,-f(-X))\colon -X\in \mathcal{D}(f)\}&\text{ set }X=-x \end{align}\]
which means \((Rf)(x)=-f(-x)=-[(-x)-5]=x+5\) is a formula for \(R(f)\).
Also \[\begin{align}M_xf&=\{M_x(x,f(x))\colon x\in \mathcal{D}(f)\}\\ &=\{(x,-f(x))\colon x\in \mathcal{D}(f)\}\\ \end{align}\]
which means \((M_xf)(x)=-f(x)=-[x-5]=-x+5\) is a formula for \(M_x(f)\).
Finally, \[\begin{align}M_yf&=\{M_y(x,f(x))\colon x\in \mathcal{D}(f)\}\\ &=\{(-x,f(x))\colon x\in \mathcal{D}(f)\}\\ &=\{(X,f(-X))\colon -X\in \mathcal{D}(f)\}&\text{ set }X=-x \end{align}\]
which means \((M_yf)(x)=f(-x)=-x-5\) is a formula for \(M_y(f)\).
In the next sub section, we will discuss another kind of involution.
Inverting the Axes
Define the function \[{\rm Inv}\colon \mathbb R^2\smallsetminus\{(a,0)\colon a\in \mathbb R\} \to \mathbb R^2\smallsetminus\{(a,0)\colon a\in \mathbb R\} \quad \text{by}\quad {\rm Inv}(a,b) = \left(a, \tfrac{1}{b}\right).\]
Essentially, we reciprocate the non-zero \(y\) values, but keep the \(x\) value the same.
Let’s look at some examples.
Example 4
Sketch the function \(f\), where \[f(x) = \frac{1}{x}.\]
The function \(g(x)=x\) is positive when \(x>0\) and some values are \((1,1)\), \((2,2)\) and \((3,3)\). Do inversion to get \((1,1)\), \((2,\tfrac{1}{2})\) and \((3,\tfrac{1}{3})\) which means the \(y\) values are getting smaller
The function \(g(x)=x\) is negative when \(x<0\) and some values are \((-1,-1)\), \((-2,-2)\) and \((-3,-3)\). Do inversion to get \((-1,-1)\), \((-2,-\tfrac{1}{2})\) and \((-3,-\tfrac{1}{3})\).
Therefore, the graph of \(f\) looks like this:
In this example, pay attention to the differences in shape compared to the previous example.
Example 5
Sketch the function \(f\), where \[f(x) = \frac{1}{x^2}.\]
The function \(g(x)=x^2\) is positive when \(x>0\) and some values are \((1,1)\), \((2,4)\) and \((3,9)\). Do inversion to get \((1,1)\), \((2,\tfrac{1}{4})\) and \((3,\tfrac{1}{9})\) which means the \(y\) values are getting smaller
The function \(g(x)=x^2\) is positive when \(x<0\) and some values are \((-1,1)\), \((-2,4)\) and \((-3,9)\). Do inversion to get \((-1,1)\), \((-2,\tfrac{1}{4})\) and \((-3,\tfrac{1}{9})\) which means the \(y\) values are getting smaller
Therefore, the graph of \(f\) looks like this:
Pay attention to the differences in shape here.
Example 6
Sketch the function \(f\), where \[f(x) = \frac{1}{x^3}.\]
The function \(g(x)=x^3\) is positive when \(x>0\) and some values are \((1,1)\), \((2,8)\) and \((3,27)\). Do inversion to get \((1,1)\), \((2,\tfrac{1}{8})\) and \((3,\tfrac{1}{27})\) which means the \(y\) values are getting smaller
The function \(g(x)=x^3\) is negative when \(x<0\) and some values are \((-1,-1)\), \((-2,-8)\) and \((-3,-27)\). Do inversion to get \((-1,-1)\), \((-2,-\tfrac{1}{8})\) and \((-3,-\tfrac{1}{27})\).
Therefore, the graph of \(f\) looks like this:
Compare this example with the previous two examples.
Example 7
Sketch the function \(f\), where \[f(x) = \frac{1}{x^4}.\]
The function \(g(x)=x^4\) is positive when \(x>0\) and some values are \((1,1)\), \((2,16)\) and \((3,81)\). Do inversion to get \((1,1)\), \((2,\tfrac{1}{16})\) and \((3,\tfrac{1}{81})\) which means the \(y\) values are getting smaller
The function \(g(x)=x^4\) is positive when \(x<0\) and some values are \((-1,1)\), \((-2,16)\) and \((-3,81)\). Do inversion to get \((-1,1)\), \((-2,\tfrac{1}{16})\) and \((-3,\tfrac{1}{81})\) which means the \(y\) values are getting smaller
Therefore, the graph of \(f\) looks like this:
Based on all the examples, do you see any general patterns you can make?