FUNCTIONS – Inverse of a function

1. 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.

2. 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.

3. 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 ?

4. 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

5. 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 ?

6. 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

7. 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

8. FUNCTIONS – Inverse of a function When mapping a functions inverse just reverse the arrows…

9. FUNCTIONS – Inverse of a function When mapping a functions inverse just, reverse the arrows… ƒ ( x ) Coordinate Points ( 3 , - 3 ) ( 4 , - 5 ) ( 5 , - 1 ) ( 6 , - 7 ) 3 -1 4 -3 5 -5 6 -7

10. FUNCTIONS – Inverse of a function When mapping a functions inverse just, reverse the arrows… ƒ -1( x ) Coordinate Points ( - 3 , 3 ) ( - 5 , 4 ) ( -1 , 5 ) ( - 7 , 6 ) 3 -1 4 -3 5 -5 6 -7

11. 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’

12. 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’

13. 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’

14. 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 y 2. Switch your ‘x’ variable and your ‘y’ variable 3. Solve for ‘y’

15. 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 y 2. Switch your ‘x’ variable and your ‘y’ variable 3. Solve for ‘y’ So :

16. FUNCTIONS – Inverse of a function EXAMPLE : Find ƒ -1(x) of ƒ(x) = ( x – 3 ) 2 1. y = ( x – 3 ) 2 Steps : 1. Change f ( x ) to y 2. Switch your ‘x’ variable and your ‘y’ variable 3. Solve for ‘y’

17. 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’

18. 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 y 2. Switch your ‘x’ variable and your ‘y’ variable 3. Solve for ‘y’

19. 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 y 2. Switch your ‘x’ variable and your ‘y’ variable 3. Solve for ‘y’

20. 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 y 2. Switch your ‘x’ variable and your ‘y’ variable 3. Solve for ‘y’ So :

21. 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

22. 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 y x f (x) 0 1 -1 -3 -1 -5

23. 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 y x f (x) 0 1 -1 -3 -1 -5

24. 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 x f (x) y x y f -1(x) -3 -1 -5 0 1 -1 0 1 -1 -3 -1 -5

25. 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

26. 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 )

27. 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 …

28. 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 )

29. 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 )