Inverses & One-to-One
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Inverses & One-to-One

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Inverses & One-to-One Inverses & One-to-One Presentation Transcript

  • CHAPTER 6
    • INVERSES & ONE-to-ONE Functions
  • Inverse Functions
    • Example
    • Also, f [ g (12)] = 12. For these functions, it can be
    • shown that
    • for any value of x . These functions are inverse functions
    • of each other.
    • Only functions that are one-to-one have inverses.
    One-to-One Functions A function f is a one-to-one function if, for elements a and b from the domain of f , a  b implies f ( a )  f ( b ).
  • One-to-One Functions
    • Example Decide whether each function is one-to-one.
    • (a) (b)
    • Solution
    • (a) For this function, two different x -values produce two different y -values.
    • (b) If we choose a = 3 and b = –3, then 3  –3, but
  • Horizontal Line Test
    • Example Use the horizontal line test to determine
    • whether the graphs are graphs of one-to-one functions.
    • (a) (b)
    If every horizontal line intersects the graph of a function at no more than one point, then the function is one-to-one. Not one-to-one One-to-one
  • Inverse Functions
    • Example
    • are inverse functions of each other.
    Let f be a one-to-one function. Then, g is the inverse function of f and f is the inverse of g if
  • Finding an Equation for the Inverse Function
    • .
    Finding the Equation of the Inverse of y = f ( x ) For a one-to-one function f defined by an equation y = f ( x ) , find the defining equation of the inverse as follows. ( Any restrictions on x and y should be considered.) 1. Interchange x and y. 2. Solve for y. 3 . Replace y with f -1 ( x ).
  • Example of Finding f -1 ( x )
    • Example Find the inverse, if it exists, of
    • Solution
    Write f ( x ) = y . Interchange x and y . Solve for y . Replace y with f -1 ( x ).
  • The Graph of f -1 ( x )
    • f and f -1 ( x ) are inverse functions, and f ( a ) = b for real numbers a and b . Then f -1 ( b ) = a .
    • If the point ( a , b ) is on the graph of f , then the point ( b , a ) is on the graph of f -1 .
    If a function is one-to-one, the graph of its inverse f -1 ( x ) is a reflection of the graph of f across the line y = x .
  • Finding the Inverse of a Function with a Restricted Domain
    • Example Let
    • Solution Notice that the domain of f is restricted
    • to [ – 5,  ), and its range is [0,  ). It is one-to-one and
    • thus has an inverse.
    • The range of f is the domain of f -1 , so its inverse is
  • Important Facts About Inverses
    • If f is one-to-one, then f -1 exists.
    • The domain of f is the range of f -1 , and the range of f is the domain of f -1 .
    • If the point ( a , b ) is on the graph of f , then the point ( b , a ) is on the graph of f -1 , so the graphs of f and f -1 are reflections of each other across the line y = x .
  • Application of Inverse Functions
    • Example Use the one-to-one function f ( x ) = 3 x + 1 and the
    • numerical values in the table to code the message BE VERY CAREFUL.
    • A 1 F 6 K 11 P 16 U 21
    • B 2 G 7 L 12 Q 17 V 22
    • C 3 H 8 M 13 R 18 W 23
    • D 4 I 9 N 14 S 19 X 24
    • E 5 J 10 O 15 T 20 Y 25
    • Z 26
    • Solution BE VERY CAREFUL would be encoded as
    • 7 16 67 16 55 76 10 4 55 16 19 64 37
    • because B corresponds to 2, and f (2) = 3(2) + 1 = 7,
    • and so on.