Inverses & One-to-One

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Inverses & One-to-One

  1. 1. CHAPTER 6 <ul><li>INVERSES & ONE-to-ONE Functions </li></ul>
  2. 2. Inverse Functions <ul><li>Example </li></ul><ul><li>Also, f [ g (12)] = 12. For these functions, it can be </li></ul><ul><li>shown that </li></ul><ul><li>for any value of x . These functions are inverse functions </li></ul><ul><li>of each other. </li></ul>
  3. 3. <ul><li>Only functions that are one-to-one have inverses. </li></ul>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 ).
  4. 4. One-to-One Functions <ul><li>Example Decide whether each function is one-to-one. </li></ul><ul><li>(a) (b) </li></ul><ul><li>Solution </li></ul><ul><li>(a) For this function, two different x -values produce two different y -values. </li></ul><ul><li>(b) If we choose a = 3 and b = –3, then 3  –3, but </li></ul>
  5. 5. Horizontal Line Test <ul><li>Example Use the horizontal line test to determine </li></ul><ul><li>whether the graphs are graphs of one-to-one functions. </li></ul><ul><li>(a) (b) </li></ul>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
  6. 6. Inverse Functions <ul><li>Example </li></ul><ul><li>are inverse functions of each other. </li></ul>Let f be a one-to-one function. Then, g is the inverse function of f and f is the inverse of g if
  7. 7. Finding an Equation for the Inverse Function <ul><li>. </li></ul>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 ).
  8. 8. Example of Finding f -1 ( x ) <ul><li>Example Find the inverse, if it exists, of </li></ul><ul><li>Solution </li></ul>Write f ( x ) = y . Interchange x and y . Solve for y . Replace y with f -1 ( x ).
  9. 9. The Graph of f -1 ( x ) <ul><li>f and f -1 ( x ) are inverse functions, and f ( a ) = b for real numbers a and b . Then f -1 ( b ) = a . </li></ul><ul><li>If the point ( a , b ) is on the graph of f , then the point ( b , a ) is on the graph of f -1 . </li></ul>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 .
  10. 10. Finding the Inverse of a Function with a Restricted Domain <ul><li>Example Let </li></ul><ul><li>Solution Notice that the domain of f is restricted </li></ul><ul><li>to [ – 5,  ), and its range is [0,  ). It is one-to-one and </li></ul><ul><li>thus has an inverse. </li></ul><ul><li>The range of f is the domain of f -1 , so its inverse is </li></ul>
  11. 11. Important Facts About Inverses <ul><li>If f is one-to-one, then f -1 exists. </li></ul><ul><li>The domain of f is the range of f -1 , and the range of f is the domain of f -1 . </li></ul><ul><li>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 . </li></ul>
  12. 12. Application of Inverse Functions <ul><li>Example Use the one-to-one function f ( x ) = 3 x + 1 and the </li></ul><ul><li>numerical values in the table to code the message BE VERY CAREFUL. </li></ul><ul><li>A 1 F 6 K 11 P 16 U 21 </li></ul><ul><li>B 2 G 7 L 12 Q 17 V 22 </li></ul><ul><li>C 3 H 8 M 13 R 18 W 23 </li></ul><ul><li>D 4 I 9 N 14 S 19 X 24 </li></ul><ul><li>E 5 J 10 O 15 T 20 Y 25 </li></ul><ul><li>Z 26 </li></ul><ul><li>Solution BE VERY CAREFUL would be encoded as </li></ul><ul><li>7 16 67 16 55 76 10 4 55 16 19 64 37 </li></ul><ul><li>because B corresponds to 2, and f (2) = 3(2) + 1 = 7, </li></ul><ul><li>and so on. </li></ul>

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