Hong Kong

Stage 4 - Stage 5

Lesson

A mathematical relation $R$`R` is a mapping of elements in one set to elements in another set. Any number of elements in the first set (usually called the *domain*) can be mapped to any number of elements in the second set (usually called the *range*).

Some relations have the property that elements in the domain are mapped to *unique* elements in the range. Such relations are called *functions*. As an example the function $y=x^2$`y`=`x`2 maps values of $x$`x` in the domain to unique values of $y$`y`. We know for instance that $3$3 is mapped to $9$9, and that $4$4 is mapped to $16$16, and that $-3$−3 is mapped to $9$9 and so on. Nowhere would we find a single number in the domain being mapped to two or more different numbers in the range.

Some functions are *one-to-one functions*. These are functions where each value of $x$`x` in the domain is mapped to a unique value of y in the range.

A function like $y=2x+3$`y`=2`x`+3 is one-to-one because every value of $x$`x` in the domain is mapped to a different value of $y$`y` in the range. The function $y=x^2$`y`=`x`2 is not one-to-one because, apart from $0$0, there are always two distinct values of $x$`x` in the domain mapped to each value of $y$`y`.

An inverse function

If, and only if, a function is one-to one, then a reverse mapping will create what is known as an * inverse function*.

To explain the idea of a reverse mapping, think about the one-to-one function $y=2x+3$`y`=2`x`+3. A reverse mapping is formed by swapping the $x$`x` and $y$`y` variables so that $x=2y+3$`x`=2`y`+3 and then making $y$`y` the subject, so that $y=\frac{1}{2}\left(x-3\right)$`y`=12(`x`−3).

Note that this new inverse function is still one-to-one as every value of $x$`x` in the domain is mapped to a different value of $y$`y` in the range.

Geometrically, swapping the $x$`x` and $y$`y` variables around ensures that the function and the inverse function are mirror images across the line $y=x$`y`=`x` as shown in the first graph.

In a reverse mapping, the range and domain are reversed as well.

For example, the one-to-one function given by $y=\frac{x-6}{x+4}$`y`=`x`−6`x`+4 has the natural domain as all real numbers excluding $x=-4$`x`=−4. The range includes all real numbers other than $y=1$`y`=1. The exclusion of $y=1$`y`=1 can be seen if we rewrite the function as $y=1-\frac{10}{x+4}$`y`=1−10`x`+4.

The inverse mapping is thus given by $x=\frac{y-6}{y+4}$`x`=`y`−6`y`+4 with the domain given by all real numbers other than $x=1$`x`=1 and the range given by all real numbers other than $y=-4$`y`=−4.

Since $x=\frac{y-6}{y+4}$`x`=`y`−6`y`+4 we can make $y$`y` the subject of the inverse function as follows:

$x$x |
$=$= | $\frac{y-6}{y+4}$y−6y+4 |

$xy+4x$xy+4x |
$=$= | $y-6$y−6 |

$xy-y$xy−y |
$=$= | $-4x-6$−4x−6 |

$y\left(x-1\right)$y(x−1) |
$=$= | $-2\left(2x+3\right)$−2(2x+3) |

$y$y |
$=$= | $-2\left(\frac{2x+3}{x-1}\right)$−2(2x+3x−1) |

Here is the graph of the function and its inverse:

Do the following graphs have inverse functions?

Yes

ANo

BYes

ANo

BYes

ANo

BYes

ANo

BYes

ANo

B

Find the inverse function of $y=x^5+1$`y`=`x`5+1.

Examine the following graph containing two lines:

Are the lines in the graph inverse functions of each other?

Yes

ANo

B