functions – inverse of a function a general rule :if ( x, y ) is a point on a function, ( y, x )...
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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…
3
4
5
6
-3
-7
-5
-1
ƒ ( x )
Coordinate Points
( 3 , - 3 )
( 4 , - 5 )
( 5 , - 1 )
( 6 , - 7 )
FUNCTIONS – Inverse of a function
When mapping a functions inverse just, reverse the arrows…
3
4
5
6
-3
-7
-5
-1
ƒ -1( x )
Coordinate Points
( - 3 , 3 )
( - 5 , 4 )
( -1 , 5 )
( - 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 y2. Switch your ‘x’ variable and your ‘y’ variable3. 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 y2. Switch your ‘x’ variable and your ‘y’ variable3. 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
3. x + 3 = 2y - added 3 to both sides
x + 3 = y - divided both sides by 2
2
Steps : 1. Change f ( x ) to y2. Switch your ‘x’ variable and your ‘y’ variable3. 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
3. x + 3 = 2y - added 3 to both sides
x + 3 = y - divided both sides by 2
2
Steps : 1. Change f ( x ) to y2. Switch your ‘x’ variable and your ‘y’ variable3. Solve for ‘y’
So : 2
3)(1
x
xf
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 y2. Switch your ‘x’ variable and your ‘y’ variable3. Solve for ‘y’
FUNCTIONS – Inverse of a function
Steps : 1. Change f ( x ) to y2. Switch your ‘x’ variable and your ‘y’ variable3. Solve for ‘y’
FUNCTIONS – Inverse of a function
Steps : 1. Change f ( x ) to y2. Switch your ‘x’ variable and your ‘y’ variable3. 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 y2. Switch your ‘x’ variable and your ‘y’ variable3. 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 - took square root of both sides
Steps : 1. Change f ( x ) to y2. Switch your ‘x’ variable and your ‘y’ variable3. 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 - took square root of both sides
√x = y – 3 - add 3 to both sides
Steps : 1. Change f ( x ) to y2. Switch your ‘x’ variable and your ‘y’ variable3. 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 - took square root of both sides
√x = y – 3 - add 3 to both sides
√x + 3 = y
Steps : 1. Change f ( x ) to y2. Switch your ‘x’ variable and your ‘y’ variable3. Solve for ‘y’
So : 3)(1 xxf
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
1
-1
-3
-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
1
-1
-3
-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
1
-1
-3
-1
-5
f -1(x) x y
-3
-1
-5
0
1
-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 )
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