# 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 $y = \log_e(x)$

- $y = \sin(x)$ and $y = \sin^{–1}(x)$

- $y = x^2$ and $y = \pm\sqrt{x}$

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The inverse of a relation can be found in 3 ways:

- 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

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

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#### 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 $y = x$

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# Inverses and Functions

Recall that a **function** is a relation that passes the vertical line test.

- for any x value there is no more than one y-value.

The inverse of a function is not necessarily a function.

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

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#### Example 2

A function is described by the rule: $f: R \rightarrow R, \;\; f(x) = (x + 2)^2$,

Find the inverse of $f(x)$

**Solution:**

… … Let $y = \big( x + 2 \big)^2$

… … domain: $x \in R$

… … range: $\big\{ y : y \geqslant 0 \big\}$

- 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**

… … **Inverse:**

… … $x = \big( y + 2 \big)^2$

… … $\big( y + 2 \big)^2 = x$

… … $y + 2 = \pm \sqrt{x}$

… … $y = \pm \sqrt{x} - 2$

… … domain: $\big\{ x : x \geqslant 0 \big\}$

so the inverse of $f(x)$ is

… … $y = \pm \sqrt{x} - 2, \quad \text{ for } \big\{ x : x \geqslant 0 \big\}$

… … notice that the inverse is **not** a function because f(x) is not one-to-one

… … the inverse fails the vertical line test

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**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 $(0, \; -2)$

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## The Inverse using the calculator

- Your Classpad calculator can
**not**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 Math1 tab, or the ACTION menu, ADVANCED submenu

**Enter:** … $\text{solve}\big( x = (y + 2)^2 , y \big)$

- Notice the two solutions.

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## The Inverse using Matrices

The matrix operation that produces a reflection across the line $y = x$ is:

… … $\left[ \begin{matrix} x' \\ y' \\ \end{matrix} \right] = \left[ \begin{matrix} 0&1 \\ 1&0 \\ \end{matrix} \right] \left[ \begin{matrix} x \\ y \\ \end{matrix} \right]$

For example, the inverse of the point $(4, \; 2)$ can be found by:

… … $\left[ \begin{matrix} x' \\ y' \\ \end{matrix} \right] = \left[ \begin{matrix} 0&1 \\ 1&0 \\ \end{matrix} \right] \left[ \begin{matrix} 4 \\ 2 \\ \end{matrix} \right]$

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… … $\left[ \begin{matrix} x' \\ y' \\ \end{matrix} \right] = \left[ \begin{matrix} 2 \\ 4 \\ \end{matrix} \right]$

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

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## 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.

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### Activity

For each of the following graphs,

- draw in the line y = x and hence sketch the inverse relation.
- Label significant points
- state domain and range of both the graph and its inverse.
- For each graph and its inverse, state if the graph is a function.

#### Example 3

**Solution**

Original | Inverse | |
---|---|---|

Domain |
$x \in [-3, \; 1]$ | $x \in [1, \; 3]$ |

Range |
$y \in [1, \; 3]$ | $y \in [-3, \; 1]$ |

Function |
no |
no |

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#### Example 4

**Solution**

Original | Inverse | |
---|---|---|

Domain |
$x \in [-2, \; 2]$ | $x \in [0, \; 2]$ |

Range |
$y \in [0, \; 2]$ | $y \in [-2, \; 2]$ |

Function |
no |
no |

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#### Example 5

**Solution**

Original | Inverse | |
---|---|---|

Domain |
$x \in R$ | $x \in R$ |

Range |
$y \in R$ | $y \in R$ |

Function |
yes |
no |

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#### Example 6

**Solution**

Original | Inverse | |
---|---|---|

Domain |
$x \in R^+$ | $x \in R$ |

Range |
$y \in R$ | $y \in R^+$ |

Function |
yes |
yes |

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