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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
"Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"
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
Function Inverse Function
x2
xa
+
-
reciprocate
x1
ax
Remember the inverse function performs the reverse effect
-
+
Inverse Functions
Using the Reciprocal Function
Ex.1 f(x)= find f–1 (x)1x
To find the inverse we need a function which will change
½ back into 2 and ¼ back into 4 etc
f–1(x) = 1x
So the inverse of is 1x
1x
f(x) = and f–1(x) = 1x
1x
4
3
2
11
f(x)x
121314
Inverse Functions
Function Inverse Function
x2
xa
+
-
reciprocate
x1
ax
Remember the inverse function performs the reverse effect
-
+
reciprocate
Inverse Functions
Finding the inverse of a function
Ex.1 f:x= 2(x+3)2 find f–1 (x)
List the operations in the order applied
x To find the inverse go backwards finding the inverse of each operation
x
so f –1 (x) =
Domain x 0 as you cannot a negative number
x
32
+3 square x 2 f(x)
2 square root -3f –1 (x)
2x
2x
32
x
Inverse Functions
As the original x value is obtained the inverse function is correct
The result can be checked by substitution
so f(2) =
substitute this value into the inverse function f-1(x)
f-1(50) =50
3 25 3 22
f(x)= 2(x+3)2 2(2+3)2 = 50
Inverse Functions
x
Ex.2 f:x x find f -1(x)
f(x)
25
3 4x
x
1 24
3 x 5
f –1 (x)
List the operations in the order applied
Go backwards finding the inverse of each operation
3 -4 reciprocate 2 +5
-5 2 reciprocate + 4 3f–1(x)
5x2
5x5
2x
45
2 x
4
52
31
x
Inverse Functions
Checking f(2) =
Substitute x = 6 into f–1(x)
f –1 (6)
2f x 5
3x 4
( )
25 6
3 2 4
This is the original x value.
The result can be checked by substitution
1 2
43 6 5
1 24
3 x 5
=2
Inverse Functions
Consider
1,1
3)(
x
xxf
x
xxf
xxf
3
)(13
)( 11 or
Why are these the same?ANS: add up the fractions
xxf
13
)(1 3 1
1x
3
1
x
x
3 x
x
An alternative Answer
Cross and Smile
Inverse Functions
Ex.2 f:x x find f -1(x)25
3 4x
1 2 4
3 x 5 1
f –1 (x)
1 2 4 20
3 5
x
x
1 4 18
3 5
x
x
Done earlier
Cross and Smile 1 2 4
3 x 5 1
Inverse Functions
Changing the Sign
Ex.1 f:x 5 - x
To change the sign of x multiply by –1
x -1 +5 f(x)
f–1(x) -1 -5 x
inverse of -1 is
f–1(x) = (x 5) x 5 5 x
Which is the same as -1
-1
Inverse Functions
Ex xxf 34)(
The inverse is 3
41
xxf )(
x -3 +4 f(x)
inverse of -3 is
3
4
x3
4 x
-3
Inverse Functions
The previous example was for
xxf 34 )(
The inverse was 3
41 xxf
)(
Suppose we form the compound function . )(1 xff
3
344 )( 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.
f–1(4 – 3x) ))(()( xffxff 11
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
So,5
4)(1 x
xf
Solution: 1. x ,45)( xxf
Solution: 2. 0x,1
)(x
xf
So, ,1
)(1
xxf 0x
5,5
2)(
x
xxfSolution: 3.
0,52
)(1 xx
xfSo,
Solution 4. ,2)( xxf 0xSo, 21 )2()( xxf
Inverse Functions
Using Long Division to Find Inverses
As x appears in 2 places it is impossible to go forwards
and backwards using the order of operations.
Do long division.x 2 x
So now x appears in
only 1 place.
Ex.1 f:x x-2x
x 2
x + 2 x
1
x+2
-2
x+2
2–
Inverse Functions
f: x2
1-x+2
x
22
x 1
f –1(x) = 2
x 1
x 1
2
x – 1
22
x 1 f –1(x) =
2 2x 1 1
2 2x 2x 1
2x x 1
x 1
List the operations in the order applied
Go backwards finding the inverse of each operation
Simplify f–1
(x)
+2 recip -2 +1 f(x)
Inverse Functions
SUMMARYTo find an inverse
function:
•Write the given function as a flow chart.
•Reverse all the steps of the flow chart.
Inverse Functions
The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.
Inverse Functions
is an example of a many-to-one function
xy sin
One-to-one and many-to-one functions
is an example of a one-to-one function
13 xy
xy sin13 xy
Consider the following graphs
and
Inverse Functions
x 2 4x2 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
xf
Notice that we start with x.
Check: e.g. If )(f,5x 5 14
)(1f 14 52
414 4)5(2
Inverse FunctionsThe 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:
x y
3
4 xy
So,3
4)(1 x
xf
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
e.g. 3 Find the inverse of 1,1
32)(
xx
xxf
Solution:Rearrange: 32)1( yMultiply by x – 1
:Remove brackets :
32 yyCollect x terms on one side: 32 yyRemove the common factor: 3)2( yy
x x
1
32
x
xy
x x
x x
x
Swap x and y:
Divide by ( y – 2):2
3
x yy
So, ,2
3)(1
x
xxf 2x
2
3
x
yx
Let y = the function: