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Warm Up

1. g. (. x. ). =. f. (. x. ). =. x. x. +. 2. f. o. g. Suppose. and. . Find. 1. Warm Up. Suppose that and find and. 2. 1. g. (. x. ). =. f. (. x. ). =. x. x. +. 2. f. o. g. Suppose. and. . Find.

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Warm Up

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  1. 1 g ( x ) = f ( x ) = x x + 2 f o g Suppose and . Find 1 Warm Up . Suppose that and find and 2

  2. 1 g ( x ) = f ( x ) = x x + 2 f o g Suppose and . Find .

  3. Suppose that andfind

  4. Suppose that andfind

  5. REVIEW Perform the indicated operation. Addition: Subtraction: Operations on Functions Multiplication: Division: Composition:

  6. Inverse Functions Section 7.8 in text

  7. Many (not ALL) actions are reversible • That is, they undo or cancel each other • A closed door can be opened • An open door can be closed • $100 can be withdrawn from a savings account • $100 can be deposited into a savings account

  8. NOT all actions are reversible • Some actions can not be undone • Explosions • Weather

  9. Mathematically, this basic concept of reversing a calculation and arriving at an original result is associated with an INVERSE.

  10. Actions and their inverses occur in everyday life • Climbing up a ladder • Inverse: Climbing down a ladder • Opening the door and turning on the lights • Inverse: Turning off the lights and closing the door

  11. A person opens a car door, gets in, and starts the engine. • Inverse: A person stops the engine, gets out, and closes the car door.

  12. Inverse operations can be described using functions. • Multiply x by 5 • Inverse: Divide x by 5 • Divide x by 20 and add 10 • Inverse: Subtract 10 from x and multiply by 20 • Multiply x by -2 and add 3 • Inverse: Subtract 3 from x and divide by -2

  13. Notation • To emphasize that a function is an inverse of said function, we use the same function name with a special notation. • Function, f(x) • Inverse Function of f(x) = f -1 (x)

  14. As we noted earlier, not every function has an inverse. So when does a function have an inverse? In words: Each different input produces its own different output. Graphically: Use the Horizontal Line test.

  15. Vertical Line Test on f: determines if f is a function Line Tests f Function f Not a Function Horizontal Line Test on f: determines if f -1 is a function Glencoe – Algebra 2 Chapter 7: Polynomial Functions

  16. Concept 1

  17. Numerically: interchange domain (x) and range ( f(x) ) Symbolically: Interchange x and y and solve for the new y to obtain f-1(x) How about if the function is given numerically or symbolically,how do you determine its inverse? Interchange Solve

  18. Putting It All Together with Examples • Does this table represent a function? • Does this function have an inverse? • Find the inverse

  19. Example • Does this table represent a function? • Does this function have an inverse? • Find the inverse

  20. Example • f(x) =3x +5 • Does this equation represent a function? • How do you know? • Does this function have an inverse? • How do you know? • Find the inverse • Confirm the inverse • f(x) = x3 + 1 • Does this equation represent a function? • How do you know? • Does this function have an inverse? • How do you know? • Find the inverse • Confirm the inverse

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