3.5 Notes.notebook

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February 11, 2013

Mar 1­1:27 PM

3.5 Inverse Relations

Let's check on the algebra you will need for this topic.

3.5 Notes.notebook

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February 11, 2013

Mar 1­1:29 PM

The equations of f(x)=2x+4 is the inverse equation to

g(x)=1/2x­2. Look at the points of these 2 equations and notice their line of reflection. Doesn't this make sense?

f(x) g(x)(1,6)(0,4)(­2,0)

3.5 Notes.notebook

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February 11, 2013

Mar 1­1:38 PM

To summarize;Given an equation y=f(x), swith x and y to create an inverse, thensolve for y to isolate it, and write the equation as f­1(x).

3.5 Notes.notebook

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February 11, 2013

Mar 1­1:38 PM

Write the inverse of g(x)=3x­6.Then use 2 points to see if your inverseproduces inverse points (x and y are switched)Then graph your 2 lines to see if they reflect acrossthe line y=x.

3.5 Notes.notebook

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February 11, 2013

Mar 1­1:47 PM

Question b) reminds you that to be a function one x can only result in one y value.Remember that graphs of functions pass the Vertical Line Test as a quick way to see if one x gives only one y.

Look at the results from Question c). What can you conclude??

The domain of a function will be the range of its inverse.The range of a function will be the domain of its inverse.This makes sense since domain is x and range is y and inverseswitches x and y.

3.5 Notes.notebook

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February 11, 2013

Mar 1­1:54 PM

b) _________________

____________________

____________________

c)

3.5 Notes.notebook

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February 11, 2013

Mar 1­1:56 PM

Let's practice writing inverse equations. Notice that functionsmight have inverses that are NOT functions.

3.5 Notes.notebook

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February 11, 2013

Mar 1­2:00 PM

3.5 Notes.notebook

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February 11, 2013

Mar 1­1:56 PM

If we wish to use the symbol f­1(x) we usually mean the inverse FUNCTION.But we cannot use the word function if the inversehas an x value that produces more than one y value.

We can easily tell if a function's inverse is also a functionby using the HORIZONTAL line test.

Ex) Look at the graph of f(x)=x2 and the graph of f(x)=Would their inverses also be functions.

In this case, since the root of x only exists for positive x values (or zero), then its inverse is NOT all of x2, but only the domain where x is greater than or equal to 0.

Find theinverse equation.

Find theinverse equation

3.5 Notes.notebook

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February 11, 2013

Mar 1­2:17 PM

Do these functions have inverses that are also valid functions?

3.5 Notes.notebook

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February 11, 2013

Mar 1­1:46 PM

In the example, notice that the algebra to producethe inverse resulted in the function . The domain of f(x) was greater than or equal to 0, but we had to RESTRICTthe Domain or the inverse function.

We can use this same concept in our original functions so that the inverse is still a FUNCTION.By restricting part of the domain, we can produce a functionthat passes both the VERTICAL and the HORIZONTAL linetest. This will ensure that the inverse is a function with one xvalue producing one y value.

Ex) Graph f(x) =x2 where x>0.Now solve for f­1(x) and draw its graph.Notice by restricting the domain, f(x) passes the VLT and HLT.Notice that f­1(x) passes the VLT which means it is a function.

3.5 Notes.notebook

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February 11, 2013

Mar 1­2:40 PM

3.5 Notes.notebook

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February 11, 2013

Mar 1­2:48 PM

The second example has the added difficulty of the restricted domain. This means that the inverse functionmust have the same RANGE!

To check whether 2 equations are inverses, find the inverse of oneand see if it matches the second equation.

3.5 Notes.notebook

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February 11, 2013

Mar 1­3:05 PM

3.5 Notes.notebook

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February 11, 2013

Mar 1­2:48 PM

3.5 Notes.notebook

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February 11, 2013

Mar 1­3:06 PM

What would happen at the intersection point of afunction and its inverse.

Find the inverse of f(x)=2x­4.

How would we algebraically solve for their intersection??

What do we notice?? Why does this make sense??

3.5 Notes.notebook

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February 11, 2013

Mar 1­3:09 PM

Homework; Page 243#4­10a,11,12, 13 (Challenge!)Mult. Ch. #1,2

Do #7a)c) and #11 in class.