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An introduction to inverse functions and derivatives

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- 1. Derivatives of Functions and Inverse Functions E. Alexander Burt Potomac School
- 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. y=f(x)
- 4. The function may also take the form of a set of equations
- 5. The function may also take the form of a table of values. </li></ul></ul>
- 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. y=3 x
- 8. y=5x-2 </li></ul>
- 9. Functions which are not One to One <ul><li>Many functions are not one to one functions. For example, graph the following:
- 10. y=x 2 for -2<x<2
- 11. y=sin(x) for - p <x< p
- 12. These functions can be made into one to one functions by restricting the domain.
- 13. y=x 2 for 0<x<2
- 14. y=sin(x) for - p /2<x< p /2 </li></ul>
- 15. To find the inverse: <ul><li>The function must be one to one. Restrict the domain if necessary
- 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. 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. Example: <ul><li>Consider y=x 2 </li><ul><li>We restrict the range to 0<x< ꝏ
- 19. Solve for x:
- 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. This is true of any point (a,b) on the function: (b,a) will lie on the inverse. </li></ul>
- 22. Derivatives of Inverse Functions: <ul><li>f(x) is a function and g(x) is its inverse:
- 23. f'(x) is the derivative of f(x) and g'(x) is the derivative of g(x).
- 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. Or g'(f(a))=1/m </li></ul></ul></ul>
- 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. 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. 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. Functions defined by tables: <ul><li>Consider the function: </li><ul><li>f(3) is 1, so 3,1 is on f.
- 30. Therefore 1,3 is on g
- 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. 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. 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. f(g(x))=x. g(f(x))=x also. </li></ul><li>Only a one to one function has an inverse
- 35. If the graph of the function passes through (a,b) then the graph of the inverse passes through (b,a)
- 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. 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)

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