05Ainvrelations

toc = Relations and their Inverses =

Recall:
A relation is a set of ordered pairs that can be listed, graphed or described by a rule.

The inverse of a rule is the reverse operation that "undoes" whatever the rule has done.

Examples of inverses you have already encountered are
 * y = x + 3 and y = x – 3
 * y = e x and y = log e x . . . . . . . ** (see **3B logarithms** ) **
 * y = sin(x) and y = sin –1 (x)
 * y = x 2 and y = ±sqrt(x) . . . . ** (see ** 2E Square Root Function** ) **.

The ** inverse ** of a relation can be found by:
 * swap the x and y coordinates of each ordered pair .... ** (or) **
 * reflect the graph of the relation across the line y = x .... ** (or) **.
 * interchange x and y in the rule and rearrange to make y the subject

The **__domain__** and __**range**__ of a relation are also swapped to form the range and domain of the inverse.


 * Example 1 **

Find the inverse of the graph shown here: ... ... ...


 * Solution:**


 * ** Swap the x and y coordinates of individual points **
 * ** Reflect the shape of the graph across the line y = x **

... ... ...

Inverses and Functions

 * Recall that a function has to pass the vertical line test.


 * The inverse of a function is not necessarily a function.


 * Only one-to-one functions will have an inverse which is also a function

** Example 2 **
A function is described by the rule: f: R —> R, f(x) = (x + 2) 2 , Find the inverse of f(x)


 * Solution:**

math . \qquad \qquad y = \big( x + 2 \big)^2 \\. \\ . \qquad \qquad \qquad \text{domain: } \; x \in R \\. \\ . \qquad \qquad \qquad \text{range:} \quad \big\{ y : y \geqslant 0 \big\} math


 * ** Swap the x and y in the rule then rearrange to make y the subject **
 * ** The range of f will become the domain of the inverse of f **

math . \qquad \qquad \text{Inverse:} \\.\\ . \qquad \qquad x = \big( y + 2 \big)^2 \\. \\ . \qquad \qquad \big( y + 2 \big)^2 = x \\. \\ . \qquad \qquad y + 2 = \pm \sqrt{x} \\.\\ . \qquad \qquad y = \pm \sqrt{x} - 2 math
 * ** Solution (Inverse) **

math . \qquad \qquad y = \pm \sqrt{x} - 2 \\. \\ . \qquad \qquad \qquad \text{domain:} \quad \big\{ x : x \geqslant 0 \big\} math


 * Note: **
 * ** The graph of f(x) has been reflected across the line y = x **
 * ** The coordinates of the turning point have been swapped **
 * ** (–2, 0) has become (0, –2) **


 * ** The inverse graph fails the vertical line test **
 * ** So the inverse of f(x) is __not__ a function **



The Inverse using the calculator

 * Your Classpad calculator cannot find an inverse directly
 * But if you swap the x and the y in the rule manually
 * The calculator can do the rearranging for you to make y the subject


 * For example 1 above, enter the following:
 * solve is in the ACTION menu, ADVANCED submenu


 * Enter
 * solve ( ** x ** = ( ** y ** + 2 ) ^2, ** y ** )


 * Notice the two solutions.

The Inverse using Matrices
The matrix operation that produces a reflection across the line y = x is: math \left[ \begin{matrix} x' \\ y' \\ \end{matrix} \right] = \left[ \begin{matrix} 0&1 \\ 1&0 \\ \end{matrix} \right] \left[ \begin{matrix} x \\ y \\ \end{matrix} \right] math

For example, the inverse of the point (4, 2) can be found by: math \left[ \begin{matrix} x' \\ y' \\ \end{matrix} \right] = \left[ \begin{matrix} 0&1 \\ 1&0 \\ \end{matrix} \right] \left[ \begin{matrix} 4 \\ 2 \\ \end{matrix} \right] = \left[ \begin{matrix} 2 \\ 4 \\ \end{matrix} \right] math

Since finding the inverse by swapping x and y values is trivial, we usually wouldn't use this method but this is a demonstration of what can be done using matrices.

Graphs of a Relation and its Inverse
If we start with the graph of any relation, we can sketch its inverse by reflecting it across the line y = x. Any particular points can be identified by swapping x and y coordinates.

Worksheet 5A Go to top of page flat

.