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… 3 4 5 6 -3 -7 -5 ƒ ( x ) Coordinate Points ( 3, - 3 ) ( 4, - 5 ) ( 5, - 1 ) ( 6, - 7 )

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

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 ** 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’

17. 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’

18. 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’

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 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 Steps :1. Change f ( x ) to y 2. Switch your ‘x’ variable and your ‘y’ variable 3. Solve for ‘y’

21. 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’

22. 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 :

23. 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

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

26. 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 -3 -5 f -1 (x) xy -3 -5 0 1

27. 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

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 )

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 ) ** notice where your function crosses the y = x line and plot those points …

30. 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 )

31. 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 )