Inverse Functions
By Dr. Marcia Tharp
This module covers:
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An Intuitive Idea About Finding An
Inverse Function.
Making a 1 to 1 check if a function
has an inverse
Finding the inverse of a function.
Graphing the inverse function.
Finding the Inverse of a Function
Take a look at the lists below.
What do you notice?
List 1
List 2
That’s right the x and y values swap places.
So we let the List 1 represent a function f(x).
List 2 will be the inverse of f(x) or f-1(x)
If (x,y) is an ordered pair of f, then (y,x) is an
ordered pair of the inverse f-1(x)
List 1=f(x)
List 2= f-1(x)
Looking for ordered pairs that
are swapped or reversed is one
way to intuitively find out if we
have an inverse function.
However we must be very
careful when doing this.
There is more . . .
Some functions do not have inverses.
Look at the y=x2 or f(x)= x2 and its inverse.
f(x) =x2
f-1 (x)
In the second table we see that f-1(x) has x=16 when
y= -4 and also x =16 when y=4. Since there are two y
values for the same x, f-1(x) is not a function. So
f(x)= x2 does not have an inverse.
f(x) =x2
f-1 (x)
Big Idea
For a function to have an
inverse, looking at the first
table each y-value of f must
have exactly one x value
If this happens a
function is one to one.
Question
Look at the lists below does this function
have an inverse?
F(x)
F-1(x) ??
Hint is F(x) one to one?
Does each y value of F(X) have exactly one x value?
F(x)
F-1(x) ??
Solution: F(X) is not one to one because 3
has two values associated with it 1 and –1.
This means that the second list F-1(x) is not a
function.
It is therefore not an inverse.
F(x)
F-1(x) ??
Finding out if a function is 1 to 1
is easier to see in a graph.
Lets look at the function g(x) below.
g(x)
Finding out if a function is 1 to 1
is easier to see in a graph.
Lets look at the function g(x) below.
g(x) and its graph.
Remember when we used the vertical line test to see
if we had a function. We checked each vertical line
to see it touched the points only once. If this
happened we were certain we had a function.
G(X) is a function.
Since we flip or reverse the points to get the inverse
we are going to flip the line test to horizontal to see
if g(x) is one to one. So if a horizontal line touches
g(x) only once than g(x) is one to one.
So g(x) has an inverse!
g(x)
Look at the list of ordered pairs and it’s graph.
Does this function have an inverse?
Hint:
Check to see if the function is one to one.
X
Y
4.0
-1
0
3.0
0
1
2.0
1
-1
1.0
2
3
3
2
-1.0
4
-2
-2.0
-2.0-1.0
1.0 2.0 3.0 4.0 5.0
Look at the list of ordered pairs and it’s graph.
Does this function have an inverse?
This function is one to one by the
horizontal line test. Click to see it.
X
Y
4.0
-1
0
3.0
0
1
2.0
1
-1
1.0
2
3
3
2
-1.0
4
-2
-2.0
-2.0-1.0
So it has an
inverse!
1.0 2.0 3.0 4.0 5.0
We know that we can swapthe
coordinates of a function to
find the coordinates of the
inverse.
But what if the function is
stated in rule form? How do we
find the rule for the inverse?
Remember that inverse rules undo each other.
So if f(x)=x+5 the rule for adding 5
Then f-1(x)=x-5 the rule for subtracting 5 is the
inverse.
Likewise if g(x)= 3x the rule for multiplying by 3
then g-1(x)= x the rule for dividing by 3 is the
inverse.
3
But how do we find the inverse of r(x)=2x-1 ??
Well here is a process to follow.
Example: Find the inverse of r(x) = 2x-1
1) Replace the function symbol r(x) with y.
y =2x –1
2) Solve for x.
y +1 =2x –1 +1
Undo the –1 by using the
opposite and adding +1 to
get x by itself.
y + 1 = 2x
Simplify
y + 1 = 2x
2
2
y +1 = x
2
Undo the multiplication by 2
Using the opposite operation
to divide by 2.
3) Reverse the order of the expressions around the equality.
So
becomes x = y +1
y +1 = x
2
2
4) Swap x and y just as we did with the points.
y= x +1
2
5) Replace y with r-1(x).
r-1(x). = x +1
2
This is the inverse of
r(x) = 2x –1
You can check the inverse
r-1(x) mentally by comparing
it to the original function r(x).
The operations on x in the inverse
function should be the opposite of
those in the original function.
See the next slide
Operations
Inverse Function
r-1(x)
= x +1
2
Original Function r(x) = 2x -1
Addition of 1
Division by 2
Subtraction of 1
Multiplication by 2
We have addition of 1 in the inverse and its
opposite subtraction of 1 in the original function
And we see division of 2 in the inverse and its opposite
multiplication of 2 in the original function.
We know we are on the right track because
inverse uses opposites to undo the original function.
O.K. It’s your turn!
Find the inverse of
f(x) = x-4
7
Solution:
1) Replace the function symbol f(x) with y
in
f(x) = x-4
7
y = x-4
7
2) Solve for x.
7y = (x-4)
7y + 4 = x
Multiply by 7
Add 4.
3) Reverse the order of the expressions around the equality.
x= 7y +4
4) Swap x and y just as we did with the points.
y=7x+4
5) Replace y with f-1(x).
f-1(x)=7x+4
This is the inverse of
f(x) = x-4
7
Notice that the inverse uses
opposite operations of the
original function f(x).
Now how do we make a graph of the inverse
function when given a function g(x)?
First lets swap x
and y to get the
inverse and see
where the new
points show up.
g(x)
Now let’s graph the inverse points in red.
Now draw a line
segment to connect
each point with its
inverse.
g(x)
g-1(x)
Now draw in the line y=x
What do you notice
about the distance to
the inverse?
g(x)
g-1(x)
To help draw in the line y=x
What do you notice
about the distance to
y=x from the inverse?
From the original point?
g(x)
g-1(x)
That’s right they are the same!
So the points and the inverse points are reflections in the line y=x.
As you look at the
graph you will see that
the diagonal distances
are equal.
g(x)
1.5
2
g-1(x)
1.5
3
2
3
2
2
This means we can graph the inverse function by
using the line y=x. By finding the reflection of
the point in the line y=x.
For example lets graph the inverse of y=2x+1.
y=x
1) Draw the line y=x.
1.5
2) Locate points on
1
y=2x + 1
1.5
1
3) Measure the
distance to y=x.
4) Locate the
reflected point that
is the same distance
away from y=x.
inverse
Connect the
points to
draw the
inverse.
Here are some links to interactive pages about
inverses. Try these than go on to the practice
problems listed in the file below this one in
blackboard.
Inverse Point Grapher-Put in the coordinates of a point and this
will show the point and its graph.
http://www.livemath.com/lmstorage/files/10185353169168/inverse.html
Inverse Maker-Put in a function and it will give you the inverse.
http://mss.math.vanderbilt.edu/cgi-bin/MSSAgent/~pscrooke/MSS/inversefunction.def
Good Luck!