04Hinverses

= Inverse Relations and Functions =

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

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

Examples of inverses you have already encountered are:

math . \quad \centerdot \;\; y = x+3 \quad \text{ and } \quad y = x - 3 \\.\\ . \quad \centerdot \;\; y = \sin \big( x \big) \quad \text{ and } \quad y = \sin^{-1} \big( x \big) \\.\\ . \quad \centerdot \;\; y = x^2 \quad \text{ and } \quad y = \sqrt{x} math

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


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


 * Example 1 **

Find the inverse of the following relations: ... ... ** (a) ** .. { ( 1, 5 ), ( 2, 7 ) , ( 2, 9 ) , ( 3, 5 ) } ... ... ** (b) ** .. y = 2x – 6 ... for .. x Î [ 0, 20 ) ... ... ** (c) ** .. the graph shown here: ... ... ... ... [[image:04Hgraph1a.gif width="264" height="230"]]


 * Solution:**

... ... ** (a) ** .. Inverse of { ( 1, 5 ), ( 2, 7 ) , ( 2, 9 ) , ( 3, 5 ) }


 * ** Swap the x and y coordinates of each ordered pair **

... ... ... ... { ( 5, 1 ), ( 7, 2 ) , ( 9, 2 ) , ( 5, 3 ) }

... ... ** (b) ** .. Inverse of y = 2x – 6 ... for .. x Î [ 0, 20 )


 * ** Swap x and y in the rule, then rearrange to make y the subject **

math . \qquad \qquad \text{Inverse:} \\.\\ . \qquad \qquad x = 2y - 6 \\.\\ . \qquad \qquad 2y - 6 = x \\ .\\ . \qquad \qquad 2y = x + 6 \\.\\ . \qquad \qquad y = \dfrac{x}{2} + 3 math


 * ** The domain and range also swap **

... ... ... ... domain of original: ... x Î [ 0, 20 )

... ... ... ... range of original: ... y Î [ –6, 34 )

... ... ... ... domain of inverse: ... x Î [ –6, 34 )


 * ** Solution (Inverse) **

math . \qquad \qquad y = \dfrac{x}{2} + 3 \quad \text{ for } \quad x \in \big[ -6, \; 34 \big) math

... ... ** (c) ** .. Inverse of the graph


 * ** 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 have an inverse that is also a function

Find the inverse of the function: .. f: R —> R, f(x) = ( x + 2 ) 2
 * Example 2 **


 * 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 **


 * Inverse functions using a ClassPad Calculator **
 * Your calculator cannot find the inverse of a function directly
 * But if you swap the x and y in the rule manually
 * The calculator can rearrange the rule to make y the subject


 * For the example above, enter the following
 * the solve command is in the ACTION menu, ADVANCED submenu
 * Enter
 * solve ( x = ( y + 2 ) ^2, y )

Notation of Inverse Functions


 * If the inverse of a function is a function, the inverse of f(x) is indicated using f –1 (x)
 * You are already familiar with this notation because you know that
 * the inverse of sin(x) is sin –1 (x)

For example,

math . \qquad \text{If} \quad f \big( x \big)) = 2x - 6 \\.\\ . \qquad \text{the inverse of } f \big( x \big) \text{ is} : \\.\\ . \qquad f^{-1} \big( x \big) = \dfrac{x}{2} + 3 math.