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FUNCTIONS – Inverse of a function
A general rule : If ( x , y ) is a point on a function, ( y , x ) is on
the function’s inverse.
FUNCTIONS – Inverse of a function
A general rule : If ( x , y ) is point on a function, ( y , x ) is on
the function’s inverse.
If you noticed, all that happened was x and y
switched positions.
FUNCTIONS – Inverse of a function
A general rule : If ( x , y ) is point on a function, ( y , x ) is on
the function’s inverse.
EXAMPLE :
The coordinate point ( 2 , - 4 ) is on ƒ( x ), what coordinate
point is on it’s inverse ?
FUNCTIONS – Inverse of a function
A general rule : If ( x , y ) is point on a function, ( y , x ) is on
the function’s inverse.
EXAMPLE :
The coordinate point ( 2 , - 4 ) is on ƒ( x ), what coordinate
point is on it’s inverse ?
ANSWER :
(-4,2)
- just switch x and y
FUNCTIONS – Inverse of a function
A general rule : If ( x , y ) is point on a function, ( y , x ) is on
the function’s inverse.
EXAMPLE :
The coordinate point ( -5 , 10 ) is on ƒ( x ), what coordinate
point is on it’s inverse ?
FUNCTIONS – Inverse of a function
A general rule : If ( x , y ) is point on a function, ( y , x ) is on
the function’s inverse.
EXAMPLE :
The coordinate point ( -5 , 10 ) is on ƒ( x ), what coordinate
point is on it’s inverse ?
ANSWER :
( 10 , - 5 )
- just switch x and y
FUNCTIONS – Inverse of a function
A general rule : If ( x , y ) is point on a function, ( y , x ) is on
the function’s inverse.
- The notation for an inverse function is
ƒ -1
- do not confuse this with a negative exponent
FUNCTIONS – Inverse of a function
When mapping a functions inverse just reverse the arrows…
FUNCTIONS – Inverse of a function
When mapping a functions inverse just, reverse the arrows…
ƒ(x)
3
4
-1
-3
Coordinate Points
(3,-3)
(4,-5)
5
-5
(5,-1)
6
-7
(6,-7)
FUNCTIONS – Inverse of a function
When mapping a functions inverse just, reverse the arrows…
ƒ -1( x )
3
4
-1
-3
Coordinate Points
(-3,3)
(-5,4)
5
-5
( -1 , 5 )
6
-7
(-7,6)
FUNCTIONS – Inverse of a function
So far we’ve looked at two easy ways to find inverse function values
using mapping and coordinate points.
The last method is finding the ALGEBRAIC INVERSE…
Steps : 1. Change f ( x ) to y
2. Switch your ‘x’ variable and your ‘y’ variable
3. Solve for ‘y’
FUNCTIONS – Inverse of a function
EXAMPLE : Find ƒ -1(x) of ƒ(x) = 2x – 3
1.
y = 2x - 3
Steps : 1. Change f ( x ) to y
2. Switch your ‘x’ variable
and your ‘y’ variable
3. Solve for ‘y’
FUNCTIONS – Inverse of a function
EXAMPLE : Find ƒ -1(x) of ƒ(x) = 2x – 3
1.
y = 2x – 3
2.
x = 2y – 3
Steps : 1. Change f ( x ) to y
2. Switch your ‘x’ variable
and your ‘y’ variable
3. Solve for ‘y’
FUNCTIONS – Inverse of a function
EXAMPLE : Find ƒ -1(x) of ƒ(x) = 2x – 3
Steps : 1. Change f ( x ) to y
2. Switch your ‘x’ variable
and your ‘y’ variable
3. Solve for ‘y’
1.
y = 2x – 3
2.
x = 2y – 3
3.
x + 3 = 2y
- added 3 to both sides
x+3 =y
- divided both sides by 2
2
FUNCTIONS – Inverse of a function
EXAMPLE : Find ƒ -1(x) of ƒ(x) = 2x – 3
Steps : 1. Change f ( x ) to y
2. Switch your ‘x’ variable
and your ‘y’ variable
3. Solve for ‘y’
1.
y = 2x – 3
2.
x = 2y – 3
3.
x + 3 = 2y
- added 3 to both sides
x+3 =y
- divided both sides by 2
2
So :
x3
f ( x) 
2
1
FUNCTIONS – Inverse of a function
EXAMPLE : Find ƒ -1(x) of ƒ(x) = ( x – 3 ) 2
** be careful here…parabolas are not one
to one. The only way to find an inverse is
to define a domain of the original function
that is one to one.
Steps : 1. Change f ( x ) to y
2. Switch your ‘x’ variable
and your ‘y’ variable
3. Solve for ‘y’
FUNCTIONS – Inverse of a function
Steps : 1. Change f ( x ) to y
2. Switch your ‘x’ variable
and your ‘y’ variable
3. Solve for ‘y’
FUNCTIONS – Inverse of a function
Steps : 1. Change f ( x ) to y
2. Switch your ‘x’ variable
and your ‘y’ variable
3. Solve for ‘y’
FUNCTIONS – Inverse of a function
EXAMPLE : Find ƒ -1(x) of ƒ(x) = ( x – 3 ) 2
1.
y=(x–3)2
2.
x=(y–3)2
Steps : 1. Change f ( x ) to y
2. Switch your ‘x’ variable
and your ‘y’ variable
3. Solve for ‘y’
FUNCTIONS – Inverse of a function
EXAMPLE : Find ƒ -1(x) of ƒ(x) = ( x – 3 ) 2
1.
y=(x–3)2
2.
x=(y–3)2
3.
√x = √ ( y – 3 ) 2
Steps : 1. Change f ( x ) to y
2. Switch your ‘x’ variable
and your ‘y’ variable
3. Solve for ‘y’
- took square root of both sides
FUNCTIONS – Inverse of a function
EXAMPLE : Find ƒ -1(x) of ƒ(x) = ( x – 3 ) 2
Steps : 1. Change f ( x ) to y
2. Switch your ‘x’ variable
and your ‘y’ variable
3. Solve for ‘y’
1.
y=(x–3)2
2.
x=(y–3)2
3.
√x = √ ( y – 3 ) 2
- took square root of both sides
√x = y – 3
- add 3 to both sides
FUNCTIONS – Inverse of a function
EXAMPLE : Find ƒ -1(x) of ƒ(x) = ( x – 3 ) 2
Steps : 1. Change f ( x ) to y
2. Switch your ‘x’ variable
and your ‘y’ variable
3. Solve for ‘y’
1.
y=(x–3)2
2.
x=(y–3)2
3.
√x = √ ( y – 3 ) 2
- took square root of both sides
√x = y – 3
- add 3 to both sides
√x + 3 = y
So :
f 1 ( x)  x  3
FUNCTIONS – Inverse of a function
GRAPHING INVERSE FUNCTIONS
STEPS :
1. Graph the given function using an ( x , y ) table
- if the graph is already shown, pick some points
2. Graph the y = x line ( line of symmetry )
3. Change your ( x , y ) points to ( y , x ) and graph them
4. Draw your function
GRAPHING INVERSE FUNCTIONS
STEPS :
1. Graph the given function using an ( x , y ) table
- if the graph is already shown, pick some points
2. Graph the y = x line ( line of symmetry )
3. Change your ( x , y ) points to ( y , x ) and graph them
4. Draw your function
EXAMPLE : Graph ƒ -1(x) if ƒ(x) = 2x - 3
f (x)
x y
0 -3
1 -1
-1 -5
GRAPHING INVERSE FUNCTIONS
STEPS :
1. Graph the given function using an ( x , y ) table
- if the graph is already shown, pick some points
2. Graph the y = x line ( line of symmetry )
3. Change your ( x , y ) points to ( y , x ) and graph them
4. Draw your function
EXAMPLE : Graph ƒ -1(x) if ƒ(x) = 2x - 3
f (x)
x y
0 -3
1 -1
-1 -5
GRAPHING INVERSE FUNCTIONS
STEPS :
1. Graph the given function using an ( x , y ) table
- if the graph is already shown, pick some points
2. Graph the y = x line ( line of symmetry )
3. Change your ( x , y ) points to ( y , x ) and graph them
4. Draw your function
EXAMPLE : Graph ƒ -1(x) if ƒ(x) = 2x - 3
f (x)
x y
f -1(x)
x
y
0 -3
-3
0
1 -1
-1
1
-1 -5
-5 -1
GRAPHING INVERSE FUNCTIONS
STEPS :
1. Graph the given function using an ( x , y ) table
- if the graph is already shown, pick some points
2. Graph the y = x line ( line of symmetry )
3. Change your ( x , y ) points to ( y , x ) and graph them
4. Draw your function
EXAMPLE : Graph ƒ -1(x) if ƒ(x) = 2x - 3
** notice that the two functions intersect
where they cross the y = x line
- These are good points to use to help
draw you inverse function
STEPS :
1. Graph the given function using an ( x , y ) table
- if the graph is already shown, pick some points
2. Graph the y = x line ( line of symmetry )
3. Change your ( x , y ) points to ( y , x ) and graph them
4. Draw your function
Example : Graph the inverse of the given function
POINTS : ( 9 , 3 )
(1,4)
( -1 , 3 )
(-3,-7)
STEPS :
1. Graph the given function using an ( x , y ) table
- if the graph is already shown, pick some points
2. Graph the y = x line ( line of symmetry )
3. Change your ( x , y ) points to ( y , x ) and graph them
4. Draw your function
Example : Graph the inverse of the given function
POINTS : ( 9 , 3 )
(1,4)
( -1 , 3 )
(-3,-7)
** notice where your function
crosses the y = x line and plot
those points …
STEPS :
1. Graph the given function using an ( x , y ) table
- if the graph is already shown, pick some points
2. Graph the y = x line ( line of symmetry )
3. Change your ( x , y ) points to ( y , x ) and graph them
4. Draw your function
Example : Graph the inverse of the given function
POINTS : ( 9 , 3 )
(1,4)
( -1 , 3 )
(-3,-7)
POINTS : ( 3 , 9 )
(4,1)
(3,-1)
(-7,-3)
STEPS :
1. Graph the given function using an ( x , y ) table
- if the graph is already shown, pick some points
2. Graph the y = x line ( line of symmetry )
3. Change your ( x , y ) points to ( y , x ) and graph them
4. Draw your function
Example : Graph the inverse of the given function
POINTS : ( 9 , 3 )
(1,4)
( -1 , 3 )
(-3,-7)
POINTS : ( 3 , 9 )
(4,1)
(3,-1)
(-7,-3)