1.6 Inverse Functions
Consider
the function 
. What does this
function do to the input number x?
First,
it multiplies the input number by 5, then it adds 2 to the resulting product.
Now,
suppose you were told that 17 was the resulting output number. What would you do to find out what the input
number was? First, you would have to subtract 2 then divide by 5 (getting 3).
That is, you would have to undo the operations that f did, and in the reverse order.
This function that subtracts two then divides by five is called the inverse of f, and is written
. It is the function
that undoes what f does. So if
, then 
.
Now
consider the function 
, the squaring function.
How would one undo the squaring function? By taking the square root?
What if the input number is - 2?
Then g (- 2) = 4. But if we take the square root of 4, we get
2, not - 2. The problem is that there
are two input numbers, - 2 and + 2, with the same output number 4. Thus, there is no way to figure out which
input number, - 2 or + 2, produced the 4, so there is no way to undo what g did.
Remember, a function must always give the same output for any given
input. It cannot sometimes give an
output of 2 when the input is 4 and other times give an output of - 2 when the
input is 4. Thus there is no inverse
function for g. This dilemma will always occur for functions
that transform two or more input numbers into the same output number.
In
order to have an inverse, a function must have only one input number corresponding to any given output number.
We
already know, from the definition of a function, that there can be only one
output number for any given input number.
If we add the requirement that there must also be only one input number for any given output number, we get
what is called a one-to-one function.
Only one-to-one functions have an inverse function.
Remember
the vertical line test.
No
vertical line intersects the graph of a function more than once.
For
one-to-one functions, we have the horizontal line test.
No horizontal line
intersects the graph of a one-to-one function more than once.
Exercise
1.6.1
Sketch the graph of 
between x = - 3 and x
= + 3. Draw a horizontal line which
intersects the y-axis at y = 4.
At the two points where the horizontal line intersects the graph, draw
vertical lines. Where do these two
vertical lines intersect the x-axis? Each of these two numbers has 4 as its
output.
Now
we will see a method of finding the equation of 
from the equation of 
which works for some,
but not all, one-to-one functions.
In the
inverse of a one-to-one function, the role of input and output numbers is
reversed, because the output number is put into the inverse function and
outputs the original input. This effect
of reversing the roles of input and output can be accomplished algebraically by
reversing the y and the x in the equation for the one-to-one
function.
For
example, suppose 
. Rewrite it in x y form as follows: 
Then
reverse the variables, replacing x
with y and y with x: 
This
equation describes the relationship between x
and y in the inverse function 
. It is not, however,
the equation for 
. To find the
equation for 
, we must solve this equation for y.
Exercise
1.6.2
Solve
the above equation for y and find the
equation for 
.
Consider
the following function: 
for 
. Now the requirement
that 
is very important,
since, without it, the function would not be one-to-one.
Exercise
1.6.3:
Sketch
the graph of
without the
restriction that
. Then draw it a second
time with the restriction. Notice that the first graph fails the
horizontal line test, but that the second graph passes the horizontal line
test.
Now
let us find the equation for 
given that 
.
First,
replace 
with y:
for 
Then
reverse the variables: 
for 
Solving
for y yields 
for 
. And since 
, we know to choose the negative square root of x to
get 
. Thus 
.
Exercise
1.6.4
Find
given that 
for 
.
There
is a very nice way to see what the graph of 
looks like merely by drawing the graph of 
.
Geometrically,
reversing x and y in the equation for 
is equivalent to switching the positions of the x and y axes in the Cartesian coordinate system by rotating the plane
180° about the line y = x.
Exercise
1.6.5
Sketch
the graphs of 
and 
from exercise 1.6.4.