Inverse Functions: Finding Inverses

2: Inverse Functions2: Inverse Functions
© Christine Crisp
““Teach A Level Maths”Teach A Level Maths”
Vol. 2: A2 Core ModulesVol. 2: A2 Core Modules

Module C3
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Functions

xy sin=13
+= xy
One-to-one and many-to-one functions
Each value of x maps to
only one value of y . . .
Consider the following graphs
Each value of x maps to
only one value of y . . .
BUT many other x values
map to that y.
and each y is mapped
from only one x.
and

Functions
One-to-one and many-to-one functions
is an example
of a one-to-one function
13
+= xy is an example
of a many-to-one
function

xy sin=

xy sin=13
+= xy
Consider the following graphs
and

Functions
Other one-to-one functions are:
x
y
1
= xy −= 4
and

Functions
Here the many-to-one
function is two-to-one
( except at one point ! )
432
−+= xxy 863 23
+−−= xxxy
Other many-to-one functions are:
This is a many-to-one
function even though it is
one-to-one in some
parts.
It’s always called many-
to-one.

Functions
1−±= xy
This is not a
function.
Functions cannot be
one-to-many.
We’ve had one-to-one and many-to-one functions,
so what about one-to-many?
One-to-many relationships do exist BUT, by
definition, these are not functions.
is one-to-many since it gives 2 values of y for
all x values greater than 1.
1,1 ≥−±= xxye.g.
So, for a function, we are sure of the y-value for
each value of x. Here we are not sure.

Inverse Functions
SUMMARY
13
+= xy

xy sin=
• A one-to-one function
maps each value of x to
one value of y and each
value of y is mapped
from only one x.
e.g. 13
+= xy
• A many-to-one
function maps each
x to one y but some
y-values will be
mapped from more
than one x.
e.g. xy sin=

Inverse Functions
42 += xy
Suppose we want to find the value of y when x = 3 if
We can easily see the answer is 10 but let’s write
out the steps using a flow chart.
We have
To find y for any x, we have
3 6 10
To find x for any y value, we reverse the process.
The reverse function “undoes” the effect of the
original and is called the inverse function.
2× 4+
x 2× 4+
x2 42 +x y=
The notation for the inverse of is)(xf )(1
xf −

Inverse Functions
2× 4+
x x2 42 +x
42)( += xxfe.g. 1 For , the flow chart is
2
4−x 2÷
4−x x4−
Reversing the process:
Finding an inverse
The inverse function is
2
4
)(1 −
=− x
xfTip: A useful check on the working is to substitute
any number into the original function and calculate y.
Then substitute this new value into the inverse. It
should give the original number.
Notice that we start with x.
Check:
5=
−
2
414
=+ 4)5(2
=−
)(1
f 14
14e.g. If ,5=x 5 =)(f

Inverse Functions
The flow chart method of finding an inverse can
be slow and it doesn’t always work so we’ll now use
another method.
e.g. 1 Find the inverse of xxf 34)( −=
Solution:
xy 34 −=
Rearrange ( to find x ):
Let y = the function:
yx −= 43
Swap x and y:
3
4 −
=x
y

Inverse Functions
another method.
Solution:
xy 34 −=
yx −= 43
3
4 −
=
Swap x and y:
x
y
3
4 −
=
x
y

Inverse Functions
another method.
Solution:
xy 34 −=
yx −= 43
3
4 −
=
Swap x and y:
x
y
3
4 −
=
x
y
So,
3
4
)(1 x
xf
−
=−

Inverse Functions
e.g. 2 Find the inverse function of 1,
1
3
)( ≠
−
= x
x
xf
)(xf
Notice that the domain excludes the value of x
that would make infinite.

Inverse Functions
1
3
)( ≠
−
= x
x
xf
Solution:
1−x
3
=y
There are 2 ways to rearrange to find x:
Either:

Inverse Functions
Either:
1−x
3
=
y
1
3
)( ≠
−
= x
x
xf
1−x
Solution:
3
=y

Inverse Functions
1−x
3
=
y
1
3
)( ≠
−
= x
x
xf
1−x
Solution:
Swap x and y: 1
3
+=
x
y
1
3
+=⇒
y
x
3
=y
Either:

Inverse Functions
or: )1( −xy 3=1−x
3
=
y
1
3
)( ≠
−
= x
x
xf
1−x
Solution:
Swap x and y: 1
3
+=
x
y
1
3
+=⇒
y
x
3
=y
Either:

Inverse Functions
or: )1( −xy 3=1−x
3
=
y
1
3
)( ≠
−
= x
x
xf
1−x
Solution:
Swap x and y: 1
3
+=
x
y
1
3
+=⇒
y
x
3
=y
Swap x
and y: x
x
y
+
=
3
3=− yyx⇒
yyx += 3⇒
y
y
x
+
=
3
⇒
Either:

Inverse Functions
So, for 1,
1
3
)( ≠
−
= x
x
xf
x
x
xf
x
xf
+
=+= −− 3
)(1
3
)( 11
or
Why are these the same?
ANS: x is a common denominator in the 2nd
form

Inverse Functions
So, for 1,
1
3
)( ≠
−
= x
x
xf
x
x
xf
x
xf
+
=+= −− 3
)(1
3
)( 11
or
The domain and range are:
)(0 1
xfx −
≠ and 1≠

Inverse Functions
The 1st
example we did was for xxf 34)( −=
The inverse was
3
4
)(1 x
xf
−
=−
Suppose we form the compound
function .
)(1
xff −
== −−
))(()( 11
xffxff
3
)34(4 x−−
3
344 x+−
=
x=)(1
xff −
⇒
Can you see why this is true for all functions that
have an inverse?
ANS: The inverse undoes what the function has done.

Inverse Functions
xxffxff == −−
)()( 11
The order in which we find the compound function
of a function and its inverse makes no difference.
For all functions which have an inverse,)(xf

Inverse Functions
Exercise
Find the inverses of the following functions:
,2)( xxf −= 0≥x
2.
3. 5,
5
2
)( −≠
+
= x
x
xf
,45)( −= xxf1. ∈x 
,
1
)(
x
xf = 0≠x
4.
See if you spot
something special about
the answer to this one.
Also, for this, show
xxff =−
)(1

Inverse Functions
Rearrange:
Swap x and y:
Let 45 −= xy
xy 54 =+
x
y
=
+
5
4
y
x
=
+
5
4
Since the x-term is
positive I’m going to work
from right to left.
So,
5
4
)(1 +
=− x
xf
Solution: 1. ∈x ,45)( −= xxf

Inverse Functions
This is an example
of a self-inverse
function.
Solution: 2. 0≠x,
1
)(
x
xf =
Let
x
y
1
=
Rearrange:
y
x
1
=
Swap x and y:
x
y
1
=
So, ,
1
)(1
x
xf =−
0≠x
)()(1
xfxf =−

Inverse Functions
5,
5
2
)( −≠
+
= x
x
xfSolution: 3.
Rearrange:
Swap x and y:
Let
5
2
+
=
x
y
y
x
2
5 =+
5
2
−=
y
x
5
2
−=
x
y
0,5
2
)(1
≠−=−
x
x
xfSo,

Inverse Functions
Solution 4. ,2)( xxf −= 0≥x
Rearrange:
Swap x and y:
Let
xy −= 2
yx −= 2
2
)2( yx −=
2
)2( xy −=
So,
21
)2()( xxf −=−
211
)2(2))(()( xxffxff −−== −−
)2(2 x−−= x=

Inverse Functions
e.g. 3 Find the inverse of 1,
1
32
)( ≠
−
+
= x
x
x
xf
Solution:
Rearrange:
Multiply by x – 1 :
Careful! We are trying to find x and it appears
twice in the equation.
32)1( +=−y x x
1
32
−
+
=
x
x
yThe next example is more difficult to rearrange

Inverse Functions
32)1( +=−y x x
Careful! We are trying to find x and it appears
twice in the equation.
1
32
)( ≠
−
+
= x
x
x
xf
Solution:
Rearrange:
We must get both x-terms on one side.
1
32
−
+
=
x
x
y

Inverse Functions
x
2
3
−
+
=
y
y
x 3)2( +=− yy
32 +=− yyx x
32 +=− yyx x
32)1( +=−y x x
1
32
)( ≠
−
+
= x
x
x
xf
Solution:
Remove brackets :
Collect x terms on one side:
Remove the common factor:
Divide by ( y – 2):
1
32
−
+
=
x
x
y
Rearrange:
Swap x and y:

Inverse Functions
x
2
3
−
+
=
y
y
x 3)2( +=− yy
32 +=− yyx x
32 +=− yyx x
32)1( +=−y x x
1
32
)( ≠
−
+
= x
x
x
xf
Solution:
Remove brackets :
Collect x terms on one side:
Swap x and y:
Remove the common factor:
Divide by ( y – 2):
1
32
−
+
=
x
x
y
Rearrange:
So, ,
2
3
)(1
−
+
=−
x
x
xf 2≠x
2
3
−
+
=
x
y
x

Inverse Functions
SUMMARY
To find an inverse function:
EITHER:
• Write the given function as a flow chart.
• Reverse all the steps of the flow chart.
OR:
• Step 2: Rearrange ( to find x )
• Step 1: Let y = the function
• Step 3: Swap x and y

Inverse Functions: Finding Inverses

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