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2 7 Bzca5e Presentation Transcript

  • 1. Section 2.7 Inverse Functions
  • 2. Inverse Functions
  • 3. The function f is a set of ordered pairs, (x,y), then the changes produced by f can be “undone” by reversing components of all the ordered pairs. The resulting relation (y,x), may or may not be a function. Inverse functions have a special “undoing” relationship.
  • 4.  
  • 5. Relations, Functions & 1:1 1:1 Functions are a subset of Functions. They are special functions where for every x, there is one y, and for every y, there is one x. Relations Reminder: The definition of function is, for every x there is only one y. Functions 1:1 Functions Inverse Functions are 1:1
  • 6. 1100 1400 1000 1300 900 1200 f(x) x 1400 1100 1300 1000 1200 900 g(x) x
  • 7. Example
  • 8. Example
  • 9. Finding the Inverse of a Function
  • 10.  
  • 11. How to Find an Inverse Function
  • 12. Example Find the inverse of f(x)=7x-1
  • 13. Example
  • 14. Example
  • 15. The Horizontal Line Test And One-to-One Functions
  • 16.  
  • 17. Horizontal Line Test b and c are not one-to-one functions because they don’t pass the horizontal line test.
  • 18. Example Graph the following function and tell whether it has an inverse function or not.
  • 19. Example Graph the following function and tell whether it has an inverse function or not.
  • 20. Graphs of f and f -1
  • 21. There is a relationship between the graph of a one-to-one function, f, and its inverse f -1 . Because inverse functions have ordered pairs with the coordinates interchanged, if the point (a,b) is on the graph of f then the point (b,a) is on the graph of f -1 . The points (a,b) and (b,a) are symmetric with respect to the line y=x. Thus graph of f -1 is a reflection of the graph of f about the line y=x.
  • 22. A function and it’s inverse graphed on the same axis.
  • 23. Example If this function has an inverse function, then graph it’s inverse on the same graph.
  • 24. Example If this function has an inverse function, then graph it’s inverse on the same graph.
  • 25. Example If this function has an inverse function, then graph it’s inverse on the same graph.
  • 26. Applications of Inverse Functions The function given by f (x)=5/9x+32 converts x degrees Celsius to an equivalent temperature in degrees Fahrenheit. a. Is f a one-to-one function? Why or why not? b. Find a formula for f -1 and interpret what it calculates. F=f (x)=5/9x+32 is 1 to 1 because it is a linear function. The Celsius formula converts x degrees Fahrenheit into Celsius. Replace the f(x) with y Solve for y, subtract 32 Multiply by 9/5 on both sides
  • 27. (a) (b) (c) (d)
  • 28. (a) (b) (c) (d)