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Function

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function

1. 1. I N V E R S E FUNCTIONS
2. 2. DOMAIN OF f RANGE OF f RANGE OF f -1 DOMAIN OF f - -1 x x y y 1 TO 1 FUNCTION f(x) = y f -1 (y) = x Dom (f) =Ran ( f -1 ) Ran (f) =Dom (f -1 ) INVERSE FUNCTION
3. 3. EXAMPLE 1 <ul><li>GIVEN THAT THE FOLLOWING </li></ul><ul><li>FUNCTIONS HAS DOMAIN R. DETERMINE </li></ul><ul><li>WHETHER INVERSE FUNCTION EXIST </li></ul><ul><li>OR NOT. </li></ul><ul><li>i. ii. </li></ul><ul><li>iii. iv. </li></ul><ul><li>v. vi. </li></ul>
4. 4. <ul><li>YES ( 1 TO 1 FUNCTION ) </li></ul><ul><li>NO ( MANY TO 1 RELATION ) </li></ul><ul><li>NO ( MANY TO 1 RELATION ) </li></ul><ul><li>YES ( 1 TO 1 FUNCTION ) </li></ul><ul><li>NO ( MANY TO 1 RELATION ) </li></ul><ul><li>YES ( 1 TO 1 FUNCTION ) </li></ul>
5. 5. EXAMPLE 2 <ul><li>DETERMINE WHICH OF THE FOLLOWING </li></ul><ul><li>FUNCTION HAS INVERSE FUNCTIONS IN </li></ul><ul><li>THE SPECIFIED DOMAINS. </li></ul><ul><li>i. </li></ul><ul><li>ii. </li></ul><ul><li>iii. </li></ul><ul><li>iv. </li></ul><ul><li>v. </li></ul><ul><li> </li></ul>
6. 6. <ul><li>YES ( 1 TO 1 FUNCTION ) </li></ul><ul><li>YES ( 1 TO 1 FUNCTION ) </li></ul><ul><li>YES ( 1 TO 1 FUNCTION ) </li></ul><ul><li>YES ( 1 TO 1 FUNCTION ) </li></ul><ul><li>NO ( MANY TO 1 FUNCTION ) </li></ul>
7. 7. EXAMPLE 3 <ul><li>DETERMINE THE DOMAIN OF THE FUNCTION SO THAT AN INVERSE FUNCTION EXISTS. </li></ul>
8. 8. EXAMPLE 4 <ul><li>FIND THE INVERSE FUNCTION FOR THE </li></ul><ul><li>FOLLOWING FUNCTIONS. </li></ul><ul><li>i. </li></ul><ul><li>ii. </li></ul><ul><li>iii. </li></ul>
9. 9. <ul><li>i. </li></ul><ul><li>ii. </li></ul>
10. 10. <ul><li>iii. </li></ul>
11. 11. EXAMPLE 5 <ul><li>DETERMINE THE RANGE OF FIND </li></ul><ul><li>THE INVERSE FUNCTION FOR THE </li></ul><ul><li>FOLLOWING FUNCTIONS AND STATE ITS </li></ul><ul><li>DOMAIN AND RANGE. </li></ul><ul><li>i. </li></ul><ul><li>ii. </li></ul><ul><li>iii. </li></ul>
12. 12. <ul><li>i. </li></ul>Dom (f) =Ran ( f -1 ) Ran (f) =Dom (f -1 )
13. 13. <ul><li>ii. </li></ul>
14. 14. <ul><li>iii. </li></ul>
15. 15. <ul><li>IF IS 1-1 FUNCTION, THE GRAPHS </li></ul><ul><li>AND ARE REFLECTIONS </li></ul><ul><li>OF EACH OTHER IN THE LINE </li></ul>x y (m,n) (n,m)
16. 16. EXAMPLE 6 <ul><li>GIVEN THAT </li></ul><ul><li>FIND ITS INVERSE AND SKETCH BOTH GRAPHS IN THE SAME DIAGRAM. </li></ul>
17. 17. x y (3,-4) (-4,3) INVERSE FUNCTION