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Chapter 8 Section 2 MA1032 Data, Functions & Graphs Sidney Butler Michigan Technological University November 29, 2006 S Butler (Michigan Tech) Chapter 8 Section 2 November 29, 2006 1 / 9 Exercise Let f 1 ( x ) = x , f 2 ( x ) = 1 1

  1. Chapter 8 Section 2 MA1032 Data, Functions & Graphs Sidney Butler Michigan Technological University November 29, 2006 S Butler (Michigan Tech) Chapter 8 Section 2 November 29, 2006 1 / 9

  2. Exercise Let f 1 ( x ) = x , f 2 ( x ) = 1 1 − x , f 5 ( x ) = x − 1 1 x , f 3 ( x ) = 1 − x , f 4 ( x ) = x , and f 6 ( x ) = x − 1 . Note that x f 1 f 2 f 3 f 4 f 5 f 6 ◦ f 1 f 1 f 2 f 3 f 4 f 5 f 6 f 2 f 2 f 1 f 4 f 3 f 6 f 5 f 3 f 3 f 5 f 1 f 6 f 2 f 4 f 4 f 4 f 6 f 2 f 5 f 1 f 3 f 5 f 5 f 3 f 6 f 1 f 4 f 2 f 6 f 6 f 4 f 5 f 2 f 3 f 1 S Butler (Michigan Tech) Chapter 8 Section 2 November 29, 2006 2 / 9

  3. Inverse Definition Suppose Q = f ( t ) is a function with the property that each value of Q determines exactly one value of t . Then f has an inverse function, f − 1 and f − 1 ( Q ) = t if and only if Q = f ( t ) . If a function has an inverse, it is said to be invertible. S Butler (Michigan Tech) Chapter 8 Section 2 November 29, 2006 3 / 9

  4. Properties If y = f ( x ) is an invertible function and y = f − 1 ( x ) is its inverse, then f − 1 ( f ( x )) = x for all values of x for which f ( x ) is defined. f ( f − 1 ( x )) = x for all values of x for which f − 1 ( x ) is defined. Example 1 1 Check that g ( x ) = 1 − x − 1 and f ( x ) = 1 + 1 − x are inverses of each other. S Butler (Michigan Tech) Chapter 8 Section 2 November 29, 2006 4 / 9

  5. Finding an Inverse Example √ x Find the inverse of h ( x ) = √ x +1 . S Butler (Michigan Tech) Chapter 8 Section 2 November 29, 2006 5 / 9

  6. Non-Invertible Functions Example Does f ( x ) = x 2 have an inverse? Horizontal Line Test Domain of f − 1 = Range of f Range of f − 1 = Domain of f S Butler (Michigan Tech) Chapter 8 Section 2 November 29, 2006 6 / 9

  7. Restricting the Domain Could we make x 2 invertible? S Butler (Michigan Tech) Chapter 8 Section 2 November 29, 2006 7 / 9

  8. Summary Definition Properties Finding an Inverse Domain & Range S Butler (Michigan Tech) Chapter 8 Section 2 November 29, 2006 8 / 9

  9. Exercise Let f 1 ( x ) = x , f 2 ( x ) = 1 1 − x , f 5 ( x ) = x − 1 1 x , f 3 ( x ) = 1 − x , f 4 ( x ) = x , and f 6 ( x ) = x − 1 . Note that x f 1 f 2 f 3 f 4 f 5 f 6 ◦ f 1 f 1 f 2 f 3 f 4 f 5 f 6 f 2 f 2 f 1 f 4 f 3 f 6 f 5 f 3 f 3 f 5 f 1 f 6 f 2 f 4 f 4 f 4 f 6 f 2 f 5 f 1 f 3 f 5 f 5 f 3 f 6 f 1 f 4 f 2 f 6 f 6 f 4 f 5 f 2 f 3 f 1 Using the table above, find the following: 1 f − 1 6 2 ( f 3 ◦ f 6 ) − 1 3 F if f 2 ◦ f 5 ◦ F = f 5 . S Butler (Michigan Tech) Chapter 8 Section 2 November 29, 2006 9 / 9

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