Suppose that and find and

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Presentation transcript:

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

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

Suppose that and find

Suppose that and find

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

Inverse Functions Section 7.8 in text

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reversible That is, they undo or cancel each other A closed door can be opened An open door can be closed $100 can be withdrawn from a savings account $100 can be deposited into a savings account

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

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

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

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.

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

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)

As we noted earlier, not every function has an inverse 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.

Chapter 7: Polynomial Functions Line Tests Vertical Line Test on f: determines if f is a function 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

Concept 1

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? Symbolically: Interchange x and y and solve for the new y to obtain f-1(x) Numerically: interchange domain (x) and range ( f(x) ) Interchange Solve

Putting It All Together with Examples Does this table represent a function? Does this function have an inverse? Find the inverse x f(x) -1 -2 1 2 3 4 -6

Example Does this table represent a function? f(x) -3 10 -2 6 -1 4 1 2 3 -10 Does this table represent a function? Does this function have an inverse? Find the inverse

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