Discovering Quadratics and Composition of Inverses

DISCOVERING QUADRATICS AND COMPOSITION OF INVERSES

LEARNING TARGETS

• Determine how to find inverses for non-invertible functions

• Be able to explain what a composition of functions is

• Prove whether two functions are inverses of each other using algebra

ACTIVITY

• Each person is given a function

• You must graph it, make an x-y table and determine its inverse.

• You will then find someone who has your inverse function and compare your graph and table.

• How do you know that they are inverses of one another, state at least two pieces of evidence.

• We will do three rounds of this (7 minutes each)

QUADRATICS…

• How come the quadratic can be “undone” by the square root but the original function fails the HLT?

• Based on our understanding we cannot have an inverse for this function…

…OR CAN WE?!?!?!

QUADRATICS…

• Take the square root function and graph it.• Reflect it over the line• What do you get?

RESTRICTING THE DOMAIN

• In order to find the inverses for non-invertible functions we can use a technique that is called restricting the domain.

• By only using as our domain of the original function we can pass the HLT and create an inverse.

• Does it work both ways?

• This technique is called restricting the domain.

RESTRICTING THE DOMAIN

• This means we can actually find inverses for any function!!!!

• Lets look at the following graph and decide the inverse function…

HOW WOULD WE WRITE THE INVERSES FOR THIS FUNCTION

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

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What are the restricted domain’s?

HOW WOULD WE WRITE THE INVERSES FOR THIS FUNCTION

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

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What are the functions between these intervals?

PIECEWISE FUNCTION TABLE

Restricted Domain Function Inverse

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HOW WOULD WE WRITE THE INVERSES FOR THIS FUNCTION

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

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YOU TRY RESTRICTING THE DOMAIN

• Find the inverses for the following functions, be sure and note what interval over each inverse occurs!

COMPOSITION OF FUNCTIONS

• Using these functions:

• Find the following:

COMPOSITION OF FUNCTIONS

• The composition of functions is inserting a function into another function

• We use the following notation when asking for the composition of two or more functions:

ORDER MATTERS!!!

• When composing two or more functions the order in which we work matters!

• It’s no different than making French Fries…

• You have to cut the potato before you can fry them!!!

• You can’t fry the potato and then cut it!!!

• In other words composition of functions works like an assembly line there is only one correct direction to go!!!

THAT’S COOL, BUT WHAT DOES IT HAVE TO DO WITH INVERSES?

• Find the inverse and then compose it with the original function

• If you take the composition of a function and its inverse you are only left with

INVERSE COMPOSITION

• But what if my original function was

• Wait a second… if they are inverses of each other than the order does not matter?!?!?!

INVERSE COMPOSITION

• This is quick and unique way to find out if functions are inverses of each other. If the composition does not result in then they are not inverses.

DETERMINE IF THE FOLLOWING FUNCTIONS ARE INVERSES

• Worksheet• Learned how to restrict the domain in order to find inverses for non-invertible functions

• The composition of functions

• How to use the composition of functions to find out if two functions are inverses

What we did: Homework: