AOS1 Topic 12: Inverse Relations and Functions

An inverse relation is the inverse of a given relation obtained by interchanging or swapping the elements of each ordered pair. In other words, if \((x, y)\) is a point in a relation \(R\), then \((y, x)\) is an element in the inverse relation \(R^{-1}\).

A relation \(R\) from set \(A\) to \(B\) is a subset of the Cartesian product of \(A\) and \(B\). \(R\) is a subset of \(A \times B\). The elements of \(R\) are in the form of an ordered pair \((a, b)\) where \(a \in A\) and \(b \in B\).

The inverse relation of \(R\) is denoted by \(R^{-1}\). \(R^{-1}\) is a subset of \(B \times A\). The elements of \(R^{-1}\) are in the form of an ordered pair \((b, a)\) where \(b \in B\) and \(a \in A\).

Inverse Relation: Definition

An inverse relation is defined as the relation obtained by interchanging the elements of each ordered pair in the given relation. By swapping the inverse relation’s domain and range, we can write the inverse relation.

If \(R\) is a relation given by \((a, b)\), then its inverse is given by \((b, a)\).

If \(R\) is a relation from set \(A\) to set \(B\), then the inverse relation \(R^{-1}\) is defined from set \(B\) to set \(A\). In other words, if \((a, b) \in R\), then \((b, a) \in R^{-1}\) and vice versa.

Inverse Relation Graph

If the graph of a relation is given, its inverse can be found by reflecting it along the line \( y = x \). To plot the inverse relation graph, follow these steps:

1. On the given graph of a relation, select some points.
2. To create new points, swap the \(x\) and \(y\) coordinates of each point.
3. Draw a line connecting all of these new points to get the graph of the inverse relation.

Inverse Relation Theorem

Statement: The inverse relation theorem states that for any relation \(R\), \((R^{-1})^{-1} = R\).

Proof:

Let’s prove the inverse relation statement using the above-mentioned mathematical definition of relation and its inverse.

If \((x, y) \in R\), then \((y, x) \in R^{-1}\) and vice versa.

Let \((x, y) \in R \Leftrightarrow (y, x) \in R^{-1}\)

\(\Rightarrow (x, y) \in (R^{-1})^{-1}\)

Every element of \(R\) is in \((R^{-1})^{-1}\).

Thus, \(R \subseteq (R^{-1})^{-1}\).

Also, following the reverse steps, we get that every element in \((R^{-1})^{-1}\) is in \(R\).

Thus, \((R^{-1})^{-1} \subseteq R\).

Hence, \((R^{-1})^{-1} = R\).

Inverse of Algebraic Relation

If an algebraic form of a relation is given, such as \(R: y = ax + b\), then the steps listed below are used to find its inverse.

1. Step 1: Interchange or swap the variables \(x\) and \(y\).

For the given algebraic equation, after interchanging variables, \(x = ay + b\).

2. Step 2: Now express \(y\) in terms of \(x\).

Here, \(y = \frac{x - b}{a}\).

3. Step 3: Then write the given algebraic relation’s inverse as

\(R^{-1}: y = \frac{x - b}{a}\).

Note: Check to see if the graphs \(y = ax + b\) and \(y = \frac{x - b}{a}\) are symmetric about the line \(y = x\).

Facts about Inverse Relation

1:If \(R\) is a symmetric relation, then \(R = R^{-1}\).

2:The curves that represent a relation and its inverse on a graph are symmetric around the line \(y = x\).

3:Every function is a relation. Not every relation is a function. In a function, each input (domain element) is associated with exactly one output (range element), while in a general relation, an input can be associated with multiple outputs.

4:If the original relation has a positive slope, the inverse relation will have a negative slope, and vice versa.

Inverse Functions

If \( f \) is a one-to-one function, then for each number \( y \) in the range of \( f \), there is exactly one number \( x \) in the domain of \( f \) such that \( f(x) = y \).

Thus, if \( f \) is a one-to-one function, a new function \( f^{-1} \), called the inverse of \( f \), may be defined by:

\[ f^{-1}(x) = y \text{ if } f(y) = x, \text{ for } x \in \text{ran } f \text{ and } y \in \text{dom } f. \]

Note: The function \( f^{-1} \) is also a one-to-one function, and \( f \) is the inverse of \( f^{-1} \).

It is not difficult to see what the relation between \( f \) and \( f^{-1} \) means geometrically. The point \((x, y)\) is on the graph of \( f^{-1} \) if the point \((y, x)\) is on the graph of \( f \). Therefore, to get the graph of \( f^{-1} \) from the graph of \( f \), the graph of \( f \) is to be reflected in the line \( y = x \).

From this, the following is evident:

\(\text{dom } f^{-1} = \text{ran } f\)
\(\text{ran } f^{-1} = \text{dom } f\)

A function has an inverse function if and only if it is one-to-one. Using the notation for composition, we can write:

\(f \circ f^{-1}(x) = x\), for all \(x \in \text{dom } f^{-1}\)
\(f^{-1} \circ f(x) = x\), for all \(x \in \text{dom } f\)

Graphing inverse functions

The transformation which reflects each point in the plane in the line \( y = x \) can be described as ‘interchanging the \( x \)- and \( y \)-coordinates of each point in the plane’ and can be written as \( (x, y) \rightarrow (y, x) \). This is read as ‘the ordered pair \( (x, y) \) is mapped to the ordered pair \( (y, x) \).’

Reflecting the graph of a function in the line \( y = x \) produces the graph of its inverse relation. Note that the image in the graph below is not a function.

If the function is one-to-one, then the image is the graph of a function. (This is because, if the function satisfies the horizontal-line test, then its reflection will satisfy the vertical-line test.)