11X1 T02 08 inverse functions (2011)

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<ul><li> 1. Inverse Relations</li></ul> <p> 2. Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained byinterchanging x and y is x = f(y)e.g. y x 3 xinverse relation is x y 3 y 3. Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained byinterchanging x and y is x = f(y)e.g. y x 3 xinverse relation is x y 3 yThe domain of the relation is the range of its inverse relation 4. Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained byinterchanging x and y is x = f(y)e.g. y x 3 xinverse relation is x y 3 yThe domain of the relation is the range of its inverse relationThe range of the relation is the domain of its inverse relation 5. Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained byinterchanging x and y is x = f(y)e.g. y x 3 xinverse relation is x y 3 yThe domain of the relation is the range of its inverse relationThe range of the relation is the domain of its inverse relationA relation and its inverse relation are reflections of each other inthe line y = x. 6. Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained byinterchanging x and y is x = f(y)e.g. y x 3 xinverse relation is x y 3 yThe domain of the relation is the range of its inverse relationThe range of the relation is the domain of its inverse relationA relation and its inverse relation are reflections of each other inthe line y = x.e.g. y x 2 7. Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained byinterchanging x and y is x = f(y)e.g. y x 3 xinverse relation is x y 3 yThe domain of the relation is the range of its inverse relationThe range of the relation is the domain of its inverse relationA relation and its inverse relation are reflections of each other inthe line y = x.e.g. y x 2domain: all real xrange: y 0 8. Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained byinterchanging x and y is x = f(y)e.g. y x 3 xinverse relation is x y 3 yThe domain of the relation is the range of its inverse relationThe range of the relation is the domain of its inverse relationA relation and its inverse relation are reflections of each other inthe line y = x.e.g. y x 2 inverse relation: x y 2domain: all real xrange: y 0 9. Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained byinterchanging x and y is x = f(y)e.g. y x 3 xinverse relation is x y 3 yThe domain of the relation is the range of its inverse relationThe range of the relation is the domain of its inverse relationA relation and its inverse relation are reflections of each other inthe line y = x.e.g. y x 2 inverse relation: x y 2domain: all real xdomain: x 0range: y 0 range: all real y 10. Inverse Functions 11. Inverse FunctionsIf an inverse relation of a function, is a function, then it is calledan inverse function. 12. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.Testing For Inverse Functions(1) Use a horizontal line test 13. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.Testing For Inverse Functions(1) Use a horizontal line test OR2 When x f y is rewritten as y g x , y g x is unique. 14. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.Testing For Inverse Functions(1) Use a horizontal line test OR2 When x f y is rewritten as y g x , y g x is unique. i y x 2 y x 15. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.Testing For Inverse Functions(1) Use a horizontal line test OR2 When x f y is rewritten as y g x , y g x is unique. i y x 2 y x 16. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.Testing For Inverse Functions(1) Use a horizontal line test OR2 When x f y is rewritten as y g x , y g x is unique. i y x 2 y xOnly has aninverse relation 17. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.Testing For Inverse Functions(1) Use a horizontal line test OR2 When x f y is rewritten as y g x , y g x is unique. i y x 2 y xOnly has an ORinverse relation x y2 y x 18. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.Testing For Inverse Functions(1) Use a horizontal line test OR2 When x f y is rewritten as y g x , y g x is unique. i y x 2 y xOnly has anORinverse relationx y2y xNOT UNIQUE 19. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.Testing For Inverse Functions(1) Use a horizontal line test OR2 When x f y is rewritten as y g x , y g x is unique. i y x 2 y ii y x 3y x xOnly has anORinverse relationx y2y xNOT UNIQUE 20. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.Testing For Inverse Functions(1) Use a horizontal line test OR2 When x f y is rewritten as y g x , y g x is unique. i y x 2 y ii y x 3y x xOnly has anORinverse relationx y2y xNOT UNIQUE 21. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.Testing For Inverse Functions(1) Use a horizontal line test OR2 When x f y is rewritten as y g x , y g x is unique. i y x 2 y ii y x 3y x xOnly has anOR Has aninverse relationx y2 inverse functiony xNOT UNIQUE 22. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.Testing For Inverse Functions(1) Use a horizontal line test OR2 When x f y is rewritten as y g x , y g x is unique. i y x 2 y ii y x 3y x xOnly has anOROR Has aninverse relationx y2x y3 inverse functiony x y3 xNOT UNIQUE 23. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.Testing For Inverse Functions(1) Use a horizontal line test OR2 When x f y is rewritten as y g x , y g x is unique. i y x 2 y ii y x 3y x xOnly has anOROR Has aninverse relationx y2 x y3 inverse functiony xy3 xNOT UNIQUEUNIQUE 24. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has aninverse function), then; f 1 f x x AND f f 1 x x 25. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then;f 1 f x x AND f f 1 x xe.g. x2f x x2 26. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then;f 1 f x x AND f f 1 x xe.g. x2f x x2x2 y2yxx2 y2 27. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then;f 1 f x x AND f f 1 x xe.g. x2f x x2x2 y2yxx2 y2 y 2 x y 2 xy 2 x y 2 x 1 y 2 x 2 2x 2y 1 x 28. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then;f 1 f x x AND f f 1 x xe.g. x2 x22f x 2 x2x 2 f 1 f x x2 x2 y21 y x x 2 x2 y2 y 2 x y 2 xy 2 x y 2 x 1 y 2 x 2 2x 2y 1 x 29. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then;f 1 f x x AND f f 1 x xe.g. x2 x22f x 2 x2x 2f 1 f x x2 x2 y2 1 y x x 2 x2 y22x 4 2x 4 y 2 x y 2 x2 x2 xy 2 x y 24x x 1 y 2 x 2 4 2x 2 x y1 x 30. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then;f 1 f x x ANDf f 1 x xe.g. x2 x22 2x 2 2f x 2 x2x 21 x f 1 f x f f 1 x x2 2x 2 2 x2 y2 1 y x x 2 1 x x2 y22x 4 2x 4 y 2 x y 2 x2 x2 xy 2 x y 24x x 1 y 2 x 2 4 2x 2 x y1 x 31. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then;f 1 f x x ANDf f 1 x xe.g. x2 x22 2x 2 2f x 2 x2x 21 x f 1 f x f f 1 x x2 2x 2 2 x2 y2 1 y x x 2 1 x x2 y22x 4 2x 42x 2 2 2x y 2 x y 2 x2 x22x 2 2 2x xy 2 x y 24x 4x x 1 y 2 x 2 44 2x 2 x x y1 x 32. (ii) Draw the inverse relation y x 33. (ii) Draw the inverse relation y x 34. (ii) Draw the inverse relation y x 35. (ii) Draw the inverse relation yxExercise 2H; 1aceg, 2, 3bdf, 5ac, 6bd, 7ac, 9bde, 10adfhj</p>