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# Inverse of functions

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### Inverse of functions

1. 1. Inverse of Functions Mathematics 4 June 22, 2012Mathematics 4 () Inverse of Functions June 22, 2012 1 / 12
2. 2. One-to-one FunctionDeﬁnitionOne-to-one function A function f is one-to-one if for any a, b ∈ {domain of f }, f (a) = f (b). Its graph passes the horizontal line test. Mathematics 4 () Inverse of Functions June 22, 2012 2 / 12
3. 3. One-to-one FunctionExamples f (x) = x3 Mathematics 4 () Inverse of Functions June 22, 2012 3 / 12
4. 4. One-to-one FunctionExamples f (x) = x2 Mathematics 4 () Inverse of Functions June 22, 2012 4 / 12
5. 5. One-to-one FunctionExamples f (x) = x2 f (x) = x2 , x ≥ 0 Mathematics 4 () Inverse of Functions June 22, 2012 4 / 12
6. 6. One-to-one FunctionExamples x f (x) = −3 4 Mathematics 4 () Inverse of Functions June 22, 2012 5 / 12
7. 7. One-to-one FunctionExamples 4 f (x) = x−2 Mathematics 4 () Inverse of Functions June 22, 2012 6 / 12
8. 8. Inverse of a FunctionDeﬁnitionInverse of a Function 1 If f is a one-to-one function, and 2 g(f (x)) = x, then: g(x) is an inverse function of f (x) Mathematics 4 () Inverse of Functions June 22, 2012 7 / 12
9. 9. Inverse of a FunctionDeﬁnitionInverse of a Function 1 If f is a one-to-one function, and 2 g(f (x)) = x, then: g(x) is an inverse function of f (x) g(x) = f −1 (x) Mathematics 4 () Inverse of Functions June 22, 2012 7 / 12
10. 10. Inverse of a FunctionDiagrams f (x) Mathematics 4 () Inverse of Functions June 22, 2012 8 / 12
11. 11. Inverse of a FunctionDiagrams f (x) f −1 (x) Mathematics 4 () Inverse of Functions June 22, 2012 8 / 12
12. 12. Inverse of a FunctionDiagrams g(x) Mathematics 4 () Inverse of Functions June 22, 2012 9 / 12
13. 13. Inverse of a FunctionDiagrams g(x) Not a function Mathematics 4 () Inverse of Functions June 22, 2012 9 / 12
14. 14. Inverse of a FunctionDiagrams g(x) Not a function There is no inverse for g(x) Mathematics 4 () Inverse of Functions June 22, 2012 9 / 12
15. 15. Inverse of a FunctionSolving for the inverseTo ﬁnd the inverse of a function, interchange the x− and y− variables andisolate y.Examples: 1 f (x) = x3 + 1 Mathematics 4 () Inverse of Functions June 22, 2012 10 / 12
16. 16. Inverse of a FunctionSolving for the inverseTo ﬁnd the inverse of a function, interchange the x− and y− variables andisolate y.Examples: 1 f (x) = x3 + 1 √ 2 g(x) = 3 y Mathematics 4 () Inverse of Functions June 22, 2012 10 / 12
17. 17. Inverse of a FunctionSolving for the inverseTo ﬁnd the inverse of a function, interchange the x− and y− variables andisolate y.Examples: 1 f (x) = x3 + 1 √ 2 g(x) = 3 y 3 h(x) = 4x − 3 Mathematics 4 () Inverse of Functions June 22, 2012 10 / 12
18. 18. Inverse of a FunctionSolving for the inverseTo ﬁnd the inverse of a function, interchange the x− and y− variables andisolate y.Examples: 1 f (x) = x3 + 1 √ 2 g(x) = 3 y 3 h(x) = 4x − 3 x+3 4 F (x) = 4x + 5 Mathematics 4 () Inverse of Functions June 22, 2012 10 / 12
19. 19. Inverse of a FunctionSolving for the inverse graphicallyTo ﬁnd the inverse of a function graphically, ﬂip the graph such that thex−axis corresponds to the y−axis, and vice versa.Examples: f (x) f −1 (x) Mathematics 4 () Inverse of Functions June 22, 2012 11 / 12
20. 20. Inverse of a FunctionSolving for the inverse graphicallyTo ﬁnd the inverse of a function graphically, ﬂip the graph such that thex−axis corresponds to the y−axis, and vice versa.Examples: f (x) f −1 (x) Mathematics 4 () Inverse of Functions June 22, 2012 11 / 12
21. 21. Inverse of a FunctionSolving for the inverse graphicallyTo ﬁnd the inverse of a function graphically, ﬂip the graph such that thex−axis corresponds to the y−axis, and vice versa.Examples: f (x) f −1 (x) Mathematics 4 () Inverse of Functions June 22, 2012 11 / 12
22. 22. Inverse of a FunctionHomework 8Find f −1 . Verify that f −1 (f (x) = f (f −1 (x)) = x. Find also the domainand range of f −1 . 1 f (x) = 5x − 2 2 f (x) = 8 − x3 x−1 3 f (x) = x+2 4 f (x) = 4 − 3x 5 f (x) = x3 + 1 4 6 f (x) = − x+1 Mathematics 4 () Inverse of Functions June 22, 2012 12 / 12