Sec.
5.1 –
Inverse Functions
The word “
inverse
” also has a similar meaning to the word “
reverse
”.
Reverse can mean
to do the opposite. To go somewhere in your car you follow a certain path. To return
back home you can just follow the path backwards or in the opposite direction. Just do
the steps backwards. Where you had turned right to get to your destination, you will turn
left at that location when you come back home. And so on.
In mathematics, we have
inverse elements
for some operations. For example, in addition
we know that 3 + (-3) = 0 . By adding the inverse (
opposite)
of 3, we
cancel
the 3 and get
0. You use this technique many times in solving equations . For example in
3
+
7x
=
17,
we would like to eliminate the 3 on the left. We can do that by adding a -3 to both
sides of the equation. Since -3 + 3 equals 0, the additive identity element, the left side
becomes 0 + 7x or, of course, just 7x. So we eliminated the 3 by adding its opposite (or
its inverse).
Now in the resulting equation
7x = 14, we would like to eliminate the 7 so we just have
x. We can do this if we multiply by its inverse element (reciprocal) on both sides of the
equation.
. Since
7 1
1
1 7
⋅
=
, the multiplicative identity element, the left side becomes
1x,
or just x. And the resulting equation is x = 2.
Now an important principle here is that
if you combine a number and its inverse value,
you will get the
identity
element for that operation!!
A number plus its additive
inverse (opposite) gives you 0 and a number times its multiplicative inverse (reciprocal)
gives you 1. We will revisit this idea later.
Now in section 4.1, the authors start out by discussing inverse
relations
! Note not
functions!
In a relation, we find the inverse values by interchanging the x and y values. If you have a
set of ordered pairs, say
{(3,2) , (4,7) , (1,5)}, then the inverse of this relation would be
{(2,3) , (7,4) , (5,1)}. You simply interchange the first and second members.
If the relation is indicated by an equation, we find the inverse equation by simply
interchanging the x and y variables. So if the relation is defined as
2
2
3
y
y
x
+
- =
, then
the inverse relation would be
2
2
3
x
x
y
+
- =
.
If the relation is indicated by our Venn diagram mapping of domain values on the left to
range values on the right, then the inverse would be mapping the values on the right to
the values on the left. Again, we are simply “reversing” the values.
Functions and inverses
Now we would like to apply the idea of an inverse to
functions
!