2: Inverse Functions2: Inverse Functions
© Christine Crisp
““Teach A Level Maths”Teach A Level Maths”
Vol. 2: A2 Core Vol. 2: A2 Core ModulesModules
Module C3
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Functions
xy sin13 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 sin13 xy
Consider the following graphs
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 xySuppose 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 haveTo 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 4x2 42 x y
The notation for the inverse of is)(xf )(1 xf
Inverse Functions
2 4x x2 42 x
42)( xxfe.g. 1 For , the flow chart is
2
4x 2 4x 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:
52
414 4)5(2
)(1f 14
14e.g. If ,5x 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
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
3
4
Swap x and y:
xy
3
4 x
y
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
3
4
Swap x and y:
xy
3
4 x
y
So,3
4)(1 x
xf
Inverse Functions
e.g. 2 Find the inverse function of
1,1
3)(
x
xxf
)(xfNotice that the domain excludes the value of x that would make infinite.
Inverse Functions
e.g. 2 Find the inverse function of
1,1
3)(
x
xxf
Solution:
Let y = the function: 1x
3y
There are 2 ways to rearrange to find x:
Either:
Inverse Functions
Either: 1x3
y
e.g. 2 Find the inverse function of
1,1
3)(
x
xxf
1xThere are 2 ways to rearrange to find x:
Solution:
Let y = the function:
3y
Inverse Functions
1x3
y
e.g. 2 Find the inverse function of
1,1
3)(
x
xxf
1xThere are 2 ways to rearrange to find x:
Solution:
Let y = the function:
Swap x and y: 13
x
y
13
y
x
3y
Either:
Inverse Functions
or: )1( xy 31x
3
y
e.g. 2 Find the inverse function of
1,1
3)(
x
xxf
1xThere are 2 ways to rearrange to find x:
Solution:
Let y = the function:
Swap x and y: 13
x
y
13
y
x
3y
Either:
Inverse Functions
or: )1( xy 31x
3
y
e.g. 2 Find the inverse function of
1,1
3)(
x
xxf
1xThere are 2 ways to rearrange to find x:
Solution:
Let y = the function:
Swap x and y: 13
x
y
13
y
x
3y
Swap x and y: x
xy
3
3 yyxyyx 3
y
yx
3
Either:
Inverse Functions
So, for 1,1
3)(
x
xxf
x
xxf
xxf
3
)(13
)( 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
xxf
x
xxf
xxf
3
)(13
)( 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 xffxff3
)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 FunctionsExercise
Find the inverses of the following functions:
,2)( xxf 0x
2.
3. 5,5
2)(
x
xxf
,45)( xxf1. x
,1
)(x
xf 0x
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
xy
5
4
yx
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. 0x,1
)(x
xf
Letx
y1
Rearrange:y
x1
Swap x and y:x
y1
So, ,1
)(1
xxf 0x
)()(1 xfxf
Inverse Functions
5,5
2)(
x
xxfSolution: 3.
Rearrange:
Swap x and y:
Let 5
2
xy
yx
25
52
y
x
52
x
y
0,52
)(1 xx
xfSo,
Inverse Functions
Solution 4. ,2)( xxf 0x
Rearrange:
Swap x and y:
Let
xy 2
yx 22)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)(
xx
xxf
Solution:Rearrange:
Let y = the function:
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
xy
The 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.
e.g. 3 Find the inverse of 1,1
32)(
xx
xxf
Solution:Rearrange: Multiply by x – 1
:
We must get both x-terms on one side.
Let y = the function: 1
32
x
xy
Inverse Functions
x2
3
yy
x 3)2( yy
32 yyx x
32 yyx x
32)1( y x x
e.g. 3 Find the inverse of 1,1
32)(
xx
xxf
Solution:
Multiply by x – 1 :Remove brackets :Collect x terms on one side:Remove the common factor:Divide by ( y – 2):
Let y = the function: 1
32
x
xy
Rearrange:
Swap x and y:
Inverse Functions
x2
3
yy
x 3)2( yy
32 yyx x
32 yyx x
32)1( y x x
e.g. 3 Find the inverse of 1,1
32)(
xx
xxf
Solution:
Multiply by x – 1 :Remove brackets :Collect x terms on one side:
Swap x and y:
Remove the common factor:Divide by ( y – 2):
Let y = the function: 1
32
x
xy
Rearrange:
So, ,2
3)(1
x
xxf 2x
2
3
x
yx
Inverse Functions
SUMMARYTo 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