INVERSE
FUNCTIONS
Set X Set Y
1
2
3
4
5
2
10
8
6
4
Remember we talked about functions---
taking a set X and mapping into a Set Y
An inverse function would reverse that
process and map from SetY back into Set X
1
2
3
4
5
2
10
8
6
4
1
2
3
4
5
2
8
6
4
If we map what we get out of the function back, we won’t
always have a function going back!!!
Recall that to determine by the graph if an equation is
a function, we have the vertical line test.
If a vertical line intersects the graph of an
equation more than one time, the equation
graphed is NOT a function.
This is a function
This is NOT a
function This is a function
If the inverse is a function, each y value could only be paired
with one x. Let’s look at a couple of graphs.
Look at a y value (for
example y = 3)and see if
there is only one x value
on the graph for it.
For any y value, a horizontal
line will only intersection the
graph once so will only have
one x value
Horizontal Line Test to see if the inverse is a function.
If a horizontal line intersects the graph of an
equation more than one time, the equation
graphed is NOT a one-to-one function and will
NOT have an inverse function.
This is a
one-to-one
function
This is NOT a
one-to-one
function
This is NOT a
one-to-one
function
Horizontal Line Test
Horizontal Line Test
• Used to determine whether a function’s
inverse
inverse will be a function by seeing if the
original function passes the horizontal line
horizontal line
test
test.
• If the original function passes
passes the horizontal
line test, then its inverse is a function
inverse is a function.
• If the original function does not pass
does not pass the
horizontal line test, then its inverse is not a
inverse is not a
function
function.
Steps for Finding the Inverse of a
One-to-One Function
Replace f(x)
with y
Trade x and
y places
Solve for y
y = f -1
(x)
Find the inverse of
Replace f(x)
with y
Trade x and
y places
Solve for y
y = f -1
(x)
 
x
x
f


2
4
x
y


2
4
y
x


2
4
  4
2 
 y
x
4
2 
 xy
x
x
xy 2
4 


x
x
y



2
4
 
x
x
x
f
2
4
1 



Let’s check this by doing
1

f
f 





 



x
x
f
f
4
2
2
4
1

x
x
x 4
2
2
4



x

Yes!
 
x
x
x
f
4
2
1 


or
Ensure f(x) is one to one
first. Domain may need
to be restricted.
Find the inverse of a function :
Find the inverse of a function :
Example 1:
Example 1: y = 6x - 12
y = 6x - 12
Step 1: Switch x and y:
Step 1: Switch x and y: x = 6y - 12
x = 6y - 12
Step 2: Solve for y:
Step 2: Solve for y: x 6y  12
x 12 6y
x 12
6
y
1
6
x  2 y
Example 2:
Example 2:
Given the function :
Given the function : y = 3x
y = 3x2
2
+ 2
+ 2 find the inverse:
find the inverse:
Step 1: Switch x and y:
Step 1: Switch x and y: x = 3y
x = 3y2
2
+ 2
+ 2
Step 2: Solve for y:
Step 2: Solve for y: x 3y2
 2
x  2 3y
2
x  2
3
y2
x  2
3
y
Ex: Find an inverse of y = -3x+6.
• Steps: -switch x & y
-solve for y
y = -3x+6
x = -3y+6
x-6 = -3y
y
x



3
6
2
3
1


 x
y
Finding the Inverse
Try
2
2
x
y
x



 
1
Given ( ) 2 7
then 2 7
7
solve for x x
2
7
2
f x x
y x
y
y
f y

 
 
 

 

479573615-356355019-PPT-3-3-Graphs-of-Inverse-Functions.ppt
Review from chapter 2
Review from chapter 2
• Relation – a mapping of input values (x-values) onto
output values (y-values).
• Here are 3 ways to show the same relation.
y = x2
x y
-2 4
-1 1
0 0
1 1
Equation
Table of
values
Graph
• Inverse relation – just think: switch the x & y-values.
x = y2
x
y 
x y
4 -2
5 -1
0 0
1 1
** the inverse
of an
equation:
switch the x &
y and solve for
y. ** the
inverse of a
table: switch
the x & y.
** the inverse of a
graph: the reflection of
the original graph in the
line y = x.
4
2 
 x
y
Consider the graph of the function 4
2
)
( 
 x
x
f
The inverse function is
2
4
)
(
1 

 x
x
f
4
2 
 x
y
2
4


x
y
4
2 
 x
y
2
4


x
y
Consider the graph of the function 4
2
)
( 
 x
x
f
The inverse function is
2
4
)
(
1 

 x
x
f
An inverse function is just a rearrangement with x and y
swapped.
So the graphs just swap x and y!
)
4
,
0
(
x
)
0
,
4
(
x
)
2
,
3
( 
 x
)
3
,
2
( 

x
4
2 
 x
y
2
4


x
y
)
4
,
0
(
x
)
0
,
4
(
x
)
2
,
3
( 
 x
)
3
,
2
( 

x
is a reflection of in the line y = x
)
(
1
x
f 
)
(x
f
x
y 
What else do you notice about the graphs?
)
4
,
4
( 

x
The function and its inverse must meet on y = x
Graph f(x) and f -1
(x)
on the same graph.
Graph f(x) and f -1
(x)
on the same graph.
Let’s consider the function and compute
some values and graph them.
  3
x
x
f 
x f (x)
-2 -8
-1 -1
0 0
1 1
2 8
Is this a function? Yes
What will “undo” a cube? A cube root
  3
1
x
x
f 

This means “inverse function”
x f -1
(x)
-8 -2
-1 -1
0 0
1 1
8 2
Let’s take the
values we got out
of the function and
put them into the
inverse function
and plot them
Notice that the x and y values
traded places for the function
and its inverse.
These functions are
reflections of each other
about the line y = x
  3
x
x
f 
  3
1
x
x
f 

(2,8)
(8,2)
(-8,-2)
(-2,-8)
Graph f(x) = 3x − 2 and
using the same set of axes.
Then compare the two graphs.
Determine the domain and range of the
function
and its inverse.

f  1

x 2
3
479573615-356355019-PPT-3-3-Graphs-of-Inverse-Functions.ppt
Verify that the functions f and g are inverses of each other.
      2
;
2
,
2
2




 x
x
g
x
x
x
f
If we graph (x - 2)2
it is a parabola shifted right 2.
Is this a one-to-one function?
This would not be one-to-one
but they restricted the domain
and are only taking the
function where x is greater
than or equal to 2 so we will
have a one-to-one function.
e.g. On the same axes, sketch the graph of
and its inverse.
2
,
)
2
( 2


 x
x
y
)
0
,
2
(
)
1
,
3
(
x
y 
)
4
,
4
(
x
Solution:
)
2
,
0
(
)
3
,
1
(
Ex: f(x)=2x2
-4 Determine whether f -1
(x) is a
function, then find the inverse equation.
2
2
4
y
x


f -1
(x) is not a function.
y = 2x2
-4
x = 2y2
-4
x+4 = 2y2
2
4



x
y
2
2
1


 x
y
OR, if you fix
the tent in the
Ex: g(x)=2x3
Inverse is a function!
y=2x3
x=2y3
3
2
y
x

y
x

3
2
3
2
x
y 
OR, if you fix the
tent in the
basement…
2
4
3
x
y 
Exercise
(d) Find and write down its domain and range.
1 (a) Sketch the function where
.
)
(x
f
y 
(e) On the same axes sketch .
)
(
1
x
f 
1
)
( 2

x
x
f
)
(x
f
)
(
1
x
f 
)
(
1
x
f
y 

(c) Suggest a suitable domain for so that the inverse
function can be found.
(b) Write down the range of . )
(x
f
1
2

x
y
(a)
Solution:
0

x
( We’ll look at the other
possibility
in a minute. )
Rearrange: 2
1 x
y 

x
y 
1
Swap: y
x 
1
Let 1
2

x
y
(d) Inverse:
1
)
(
1


 
x
x
f
Domain: 1


x Range: 0

y
0

x
(c) Restricted domain:
(b) Range of :
)
(x
f
1
)
( 

x
f
1
2

x
y
Solution:
(a)
Rearrange: 2
1 x
y 

(d) Let 1
2

x
y
As before
(c)
0

x
Suppose you chose
for the domain
We now need since
x
y 

 1 0

x
(b) Range of :
)
(x
f
1
)
( 

x
f
1
2

x
y
Solution:
(a)
Swap: y
x 

 1
1
)
(
1




x
x
f
Range:
(b) 1


y
Domain: 1


x Range: 0

y
(c)
0

x
Suppose you chose
for the domain
Rearrange: 2
1 x
y 

(d) Let 1
2

x
y
As before
We now need since
x
y 

 1 0

x
Choosing
is easier!
0

x

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479573615-356355019-PPT-3-3-Graphs-of-Inverse-Functions.ppt

  • 2. Set X Set Y 1 2 3 4 5 2 10 8 6 4 Remember we talked about functions--- taking a set X and mapping into a Set Y An inverse function would reverse that process and map from SetY back into Set X 1 2 3 4 5 2 10 8 6 4
  • 3. 1 2 3 4 5 2 8 6 4 If we map what we get out of the function back, we won’t always have a function going back!!!
  • 4. Recall that to determine by the graph if an equation is a function, we have the vertical line test. If a vertical line intersects the graph of an equation more than one time, the equation graphed is NOT a function. This is a function This is NOT a function This is a function
  • 5. If the inverse is a function, each y value could only be paired with one x. Let’s look at a couple of graphs. Look at a y value (for example y = 3)and see if there is only one x value on the graph for it. For any y value, a horizontal line will only intersection the graph once so will only have one x value Horizontal Line Test to see if the inverse is a function.
  • 6. If a horizontal line intersects the graph of an equation more than one time, the equation graphed is NOT a one-to-one function and will NOT have an inverse function. This is a one-to-one function This is NOT a one-to-one function This is NOT a one-to-one function
  • 7. Horizontal Line Test Horizontal Line Test • Used to determine whether a function’s inverse inverse will be a function by seeing if the original function passes the horizontal line horizontal line test test. • If the original function passes passes the horizontal line test, then its inverse is a function inverse is a function. • If the original function does not pass does not pass the horizontal line test, then its inverse is not a inverse is not a function function.
  • 8. Steps for Finding the Inverse of a One-to-One Function Replace f(x) with y Trade x and y places Solve for y y = f -1 (x)
  • 9. Find the inverse of Replace f(x) with y Trade x and y places Solve for y y = f -1 (x)   x x f   2 4 x y   2 4 y x   2 4   4 2   y x 4 2   xy x x xy 2 4    x x y    2 4   x x x f 2 4 1     Let’s check this by doing 1  f f            x x f f 4 2 2 4 1  x x x 4 2 2 4    x  Yes!   x x x f 4 2 1    or Ensure f(x) is one to one first. Domain may need to be restricted.
  • 10. Find the inverse of a function : Find the inverse of a function : Example 1: Example 1: y = 6x - 12 y = 6x - 12 Step 1: Switch x and y: Step 1: Switch x and y: x = 6y - 12 x = 6y - 12 Step 2: Solve for y: Step 2: Solve for y: x 6y  12 x 12 6y x 12 6 y 1 6 x  2 y
  • 11. Example 2: Example 2: Given the function : Given the function : y = 3x y = 3x2 2 + 2 + 2 find the inverse: find the inverse: Step 1: Switch x and y: Step 1: Switch x and y: x = 3y x = 3y2 2 + 2 + 2 Step 2: Solve for y: Step 2: Solve for y: x 3y2  2 x  2 3y 2 x  2 3 y2 x  2 3 y
  • 12. Ex: Find an inverse of y = -3x+6. • Steps: -switch x & y -solve for y y = -3x+6 x = -3y+6 x-6 = -3y y x    3 6 2 3 1    x y
  • 13. Finding the Inverse Try 2 2 x y x      1 Given ( ) 2 7 then 2 7 7 solve for x x 2 7 2 f x x y x y y f y           
  • 15. Review from chapter 2 Review from chapter 2 • Relation – a mapping of input values (x-values) onto output values (y-values). • Here are 3 ways to show the same relation. y = x2 x y -2 4 -1 1 0 0 1 1 Equation Table of values Graph
  • 16. • Inverse relation – just think: switch the x & y-values. x = y2 x y  x y 4 -2 5 -1 0 0 1 1 ** the inverse of an equation: switch the x & y and solve for y. ** the inverse of a table: switch the x & y. ** the inverse of a graph: the reflection of the original graph in the line y = x.
  • 17. 4 2   x y Consider the graph of the function 4 2 ) (   x x f The inverse function is 2 4 ) ( 1    x x f 4 2   x y 2 4   x y
  • 18. 4 2   x y 2 4   x y Consider the graph of the function 4 2 ) (   x x f The inverse function is 2 4 ) ( 1    x x f An inverse function is just a rearrangement with x and y swapped. So the graphs just swap x and y! ) 4 , 0 ( x ) 0 , 4 ( x ) 2 , 3 (   x ) 3 , 2 (   x
  • 19. 4 2   x y 2 4   x y ) 4 , 0 ( x ) 0 , 4 ( x ) 2 , 3 (   x ) 3 , 2 (   x is a reflection of in the line y = x ) ( 1 x f  ) (x f x y  What else do you notice about the graphs? ) 4 , 4 (   x The function and its inverse must meet on y = x
  • 20. Graph f(x) and f -1 (x) on the same graph.
  • 21. Graph f(x) and f -1 (x) on the same graph.
  • 22. Let’s consider the function and compute some values and graph them.   3 x x f  x f (x) -2 -8 -1 -1 0 0 1 1 2 8 Is this a function? Yes What will “undo” a cube? A cube root   3 1 x x f   This means “inverse function” x f -1 (x) -8 -2 -1 -1 0 0 1 1 8 2 Let’s take the values we got out of the function and put them into the inverse function and plot them Notice that the x and y values traded places for the function and its inverse. These functions are reflections of each other about the line y = x   3 x x f    3 1 x x f   (2,8) (8,2) (-8,-2) (-2,-8)
  • 23. Graph f(x) = 3x − 2 and using the same set of axes. Then compare the two graphs. Determine the domain and range of the function and its inverse.  f  1  x 2 3
  • 25. Verify that the functions f and g are inverses of each other.       2 ; 2 , 2 2      x x g x x x f If we graph (x - 2)2 it is a parabola shifted right 2. Is this a one-to-one function? This would not be one-to-one but they restricted the domain and are only taking the function where x is greater than or equal to 2 so we will have a one-to-one function.
  • 26. e.g. On the same axes, sketch the graph of and its inverse. 2 , ) 2 ( 2    x x y ) 0 , 2 ( ) 1 , 3 ( x y  ) 4 , 4 ( x Solution: ) 2 , 0 ( ) 3 , 1 (
  • 27. Ex: f(x)=2x2 -4 Determine whether f -1 (x) is a function, then find the inverse equation. 2 2 4 y x   f -1 (x) is not a function. y = 2x2 -4 x = 2y2 -4 x+4 = 2y2 2 4    x y 2 2 1    x y OR, if you fix the tent in the
  • 28. Ex: g(x)=2x3 Inverse is a function! y=2x3 x=2y3 3 2 y x  y x  3 2 3 2 x y  OR, if you fix the tent in the basement… 2 4 3 x y 
  • 29. Exercise (d) Find and write down its domain and range. 1 (a) Sketch the function where . ) (x f y  (e) On the same axes sketch . ) ( 1 x f  1 ) ( 2  x x f ) (x f ) ( 1 x f  ) ( 1 x f y   (c) Suggest a suitable domain for so that the inverse function can be found. (b) Write down the range of . ) (x f
  • 30. 1 2  x y (a) Solution: 0  x ( We’ll look at the other possibility in a minute. ) Rearrange: 2 1 x y   x y  1 Swap: y x  1 Let 1 2  x y (d) Inverse: 1 ) ( 1     x x f Domain: 1   x Range: 0  y 0  x (c) Restricted domain: (b) Range of : ) (x f 1 ) (   x f
  • 31. 1 2  x y Solution: (a) Rearrange: 2 1 x y   (d) Let 1 2  x y As before (c) 0  x Suppose you chose for the domain We now need since x y    1 0  x (b) Range of : ) (x f 1 ) (   x f
  • 32. 1 2  x y Solution: (a) Swap: y x    1 1 ) ( 1     x x f Range: (b) 1   y Domain: 1   x Range: 0  y (c) 0  x Suppose you chose for the domain Rearrange: 2 1 x y   (d) Let 1 2  x y As before We now need since x y    1 0  x Choosing is easier! 0  x