Inverse Functions


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An introduction to inverse functions and derivatives

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Inverse Functions

  1. 1. Derivatives of Functions and Inverse Functions E. Alexander Burt Potomac School
  2. 2. Review of Functions <ul><li>A function is a rule which maps one set of numbers (the domain) to another set of numbers (the range) </li><ul><li>The function may take the form of an equation:
  3. 3. y=f(x)
  4. 4. The function may also take the form of a set of equations
  5. 5. The function may also take the form of a table of values. </li></ul></ul>
  6. 6. One to One functions <ul><li>In math-speak, for every value c, the function has at most one solution f(x) = c </li><ul><li>In plain English: if y=f(x) the function never produces the same y value twice. </li></ul><li>As examples, try graphing the following functions
  7. 7. y=3 x
  8. 8. y=5x-2 </li></ul>
  9. 9. Functions which are not One to One <ul><li>Many functions are not one to one functions. For example, graph the following:
  10. 10. y=x 2 for -2<x<2
  11. 11. y=sin(x) for - p <x< p
  12. 12. These functions can be made into one to one functions by restricting the domain.
  13. 13. y=x 2 for 0<x<2
  14. 14. y=sin(x) for - p /2<x< p /2 </li></ul>
  15. 15. To find the inverse: <ul><li>The function must be one to one. Restrict the domain if necessary
  16. 16. If the function is in the form of y=f(x) where f (x) is some algebraic expression, solve for x </li><ul><li>You will have a new function of the form x=g(y)
  17. 17. Now change the variables to make it “look normal” y=g(x) </li></ul><li>If the function is in the form of a table of values, interchange the columns. </li></ul>
  18. 18. Example: <ul><li>Consider y=x 2 </li><ul><li>We restrict the range to 0<x< ꝏ
  19. 19. Solve for x:
  20. 20. Swap the x and y: </li></ul><li>Because of the x and y swap, if the point 4,2 lies on the original function (it does) the point 2,4 will lie on the inverse (and it also does)
  21. 21. This is true of any point (a,b) on the function: (b,a) will lie on the inverse. </li></ul>
  22. 22. Derivatives of Inverse Functions: <ul><li>f(x) is a function and g(x) is its inverse:
  23. 23. f'(x) is the derivative of f(x) and g'(x) is the derivative of g(x).
  24. 24. Now suppose f(a)=b and f'(a)=m </li><ul><li>The slope of the inverse function is the inverse of the slope of the function. </li><ul><li>In symbols: g'(b) = 1/m
  25. 25. Or g'(f(a))=1/m </li></ul></ul></ul>
  26. 26. Example: <ul><li>Consider our old friend f(x)=x 2 for 0<x< ꝏ </li><ul><li>The derivative f'(x)=2x
  27. 27. 3,9 is a point on the function. The slope of the tangent at x=3 is 6 </li></ul><li>The inverse function is
  28. 28. The derivative of the inverse function is </li><ul><li>9,3 is a point on the inverse function. The slope of the tangent at x=9 is 1/6 </li></ul></ul>
  29. 29. Functions defined by tables: <ul><li>Consider the function: </li><ul><li>f(3) is 1, so 3,1 is on f.
  30. 30. Therefore 1,3 is on g
  31. 31. f'(1) is 2, so g'(1) is ½ </li></ul><li>Note that this table represents a one to one function. No number in the f(x) column appears more than once. </li></ul>x f(x) f'(x) 1 -2 2 3 1 1 4 3 0 7 -1 2 11 2 3
  32. 32. Functions defined graphically An example of a function (orange) and its inverse (blue). Note that for every (a,b) pair on the function, there is a corresponding (b,a) pair on the inverse. y=x g(x) f(x) (4,1.5) (1.5,4)
  33. 33. The conclusion, when all has been heard <ul><li>If f(x) and g(x) are inverse functions then </li><ul><li>If f(a)=b then g(b)=a
  34. 34. f(g(x))=x. g(f(x))=x also. </li></ul><li>Only a one to one function has an inverse
  35. 35. If the graph of the function passes through (a,b) then the graph of the inverse passes through (b,a)
  36. 36. Using the above notation, the slope of the tangent line to the function at a is the reciprocal of the slope of the tangent line to the inverse at b. </li></ul>
  37. 37. Not exactly Haiku... The slope of the line tangent to the inverse of a function at (a,b) is the inverse of the slope of the line tangent to the function at (b,a)