Algebra 2 Ch 3.4 Use Inverse Functions .notebook
Algebra 2 Ch 3.4 Use Inverse Functions .notebook
Use Inverse Functions
November 27, 2012
Use Inverse Functions When two functions are inverses, they have a special relationship.
3.4 Use Inverse Funcons •
Inverse Funcons
•
Composites of Inverse Funcons
•
Four steps to finding an inverse funcon
1. They switch domains and ranges. 2. Their composites are equal to x.
Nov 611:11 AM
Nov 611:11 AM
Use Inverse Functions
Use Inverse Functions
When two relations are inverses, they switch their domain and range. Which means they switch their xvalues and y values. Original Relation
x y
2 3
1 4
3 0
6 1
10 8
Nov 611:11 AM
Nov 611:11 AM
Use Inverse Functions
Use Inverse Functions
Original Relation
x y
2 3
1 4
3 0
6 1
10 8
4 1
0 3
1 6
8 10
Inverse Relation
x y
3 2
Nov 611:11 AM
Nov 611:11 AM
1
Algebra 2 Ch 3.4 Use Inverse Functions .notebook
Use Inverse Functions When relations are inverses, they are reflective over the line y=x.
November 27, 2012
Use Inverse Functions If you are given an equation, you can find its inverse.
The inverse of
is
Nov 611:11 AM
Use Inverse Functions
Nov 611:11 AM
Use Inverse Functions Steps for Finding the Inverse of a Function
Find for .
1. Replace 2. Switch
We will use a four step process to find the inverse of a function.
and
3. Solve for 4. Replace
Nov 611:11 AM
Use Inverse Functions 1. Replace
with
with
Nov 611:11 AM
Use Inverse Functions
with 2. Switch
Nov 611:11 AM
and
Nov 611:11 AM
2
Algebra 2 Ch 3.4 Use Inverse Functions .notebook
Use Inverse Functions
November 27, 2012
Use Inverse Functions
3. Solve for 4. Replace
Nov 611:11 AM
Use Inverse Functions
with
Nov 611:11 AM
Use Inverse Functions
Further Simplification of the Problem
Find for
Nov 611:11 AM
Use Inverse Functions 1. Replace
with
Nov 611:11 AM
Nov 611:11 AM
Use Inverse Functions 2. Switch
and
Nov 611:11 AM
3
Algebra 2 Ch 3.4 Use Inverse Functions .notebook
Use Inverse Functions 3. Solve for
Use Inverse Functions 3. Solve for
Nov 611:11 AM
Use Inverse Functions 4. Replace
November 27, 2012
Nov 611:11 AM
Use Inverse Functions
with
Find for
Nov 611:11 AM
Use Inverse Functions 1. Replace
with
Nov 611:11 AM
Nov 611:11 AM
Use Inverse Functions 2. Switch
and
Nov 611:11 AM
4
Algebra 2 Ch 3.4 Use Inverse Functions .notebook
Use Inverse Functions 3. Solve for
Use Inverse Functions 3. Solve for
Nov 611:11 AM
Use Inverse Functions 4. Replace
November 27, 2012
with
Nov 611:11 AM
Use Inverse Functions Example #3 from the book on page 191. An elastic band's resistance R in pounds can be modeled by:
where L is the total length of the stretched rubber band in inches. • Find the inverse of the model. • Use the inverse function to find the length at which the band provides 19 pounds of resistance.
Nov 611:11 AM
Use Inverse Functions 1. Skip step #1
Nov 611:11 AM
Use Inverse Functions 3. Solve for
2. Skip step #2 3. Solve for
Nov 611:11 AM
Nov 611:11 AM
5
Algebra 2 Ch 3.4 Use Inverse Functions .notebook
Use Inverse Functions
November 27, 2012
Use Inverse Functions
4. Skip step #4
Now solve for The rubber band stretches 64 inches when 19 pounds of force is applied to it.
Nov 611:11 AM
Use Inverse Functions
What happens when you make a composite out of inverse functions?
Nov 611:11 AM
Use Inverse Functions
Find:
Nov 611:11 AM
Nov 611:11 AM
Use Inverse Functions The two functions below are inverses, find their composites, f(g(x)) and g(f(x)).
Nov 611:11 AM
Use Inverse Functions
Find:
Nov 611:11 AM
6
Algebra 2 Ch 3.4 Use Inverse Functions .notebook
Use Inverse Functions
November 27, 2012
Use Inverse Functions
The composite of inverses is always equal to x.
Nov 611:11 AM
Use Inverse Functions
Nov 611:11 AM
Use Inverse Functions When two functions are inverses, the domain of the inner function becomes the range of the outer function.
Nov 611:11 AM
Use Inverse Functions
Nov 611:11 AM
Use Inverse Functions
Are and inverses?
You need to find the composites of the functions to prove if they are inverses or not.
Nov 611:11 AM
Nov 611:11 AM
7
Algebra 2 Ch 3.4 Use Inverse Functions .notebook
Use Inverse Functions
November 27, 2012
Use Inverse Functions
They are not inverses, because the composite is not equal to x. You do not have to find the other composite.
Nov 611:11 AM
Use Inverse Functions
Nov 611:11 AM
Use Inverse Functions
If
Find: This just proved what we already knew.
Nov 611:11 AM
Use Inverse Functions
Nov 611:11 AM
Nov 611:11 AM
Use Inverse Functions
Nov 611:11 AM
8
Algebra 2 Ch 3.4 Use Inverse Functions .notebook
Use Inverse Functions Since
Then
November 27, 2012
Use Inverse Functions
Horizontal Line Test is its own inverse.
Nov 611:11 AM
Use Inverse Functions
Nov 611:11 AM
Use Inverse Functions
Nov 611:11 AM
The horizontal line test will tell you if the inverse of a function is itself a function.
Nov 611:11 AM
Use Inverse Functions
Nov 611:11 AM
Use Inverse Functions
Nov 611:11 AM
9
Algebra 2 Ch 3.4 Use Inverse Functions .notebook
November 27, 2012
Use Inverse Functions 3.4 Use Inverse Funcons •
Inverse Funcons
•
Composites of Inverse Funcons
•
Four steps to finding an inverse funcon
Nov 611:11 AM
10
Attachments
Algebra 2 Ch 3.4 Inverse Functions, inverse of itself.gsp
Use Inverse Functions
November 27, 2012
Use Inverse Functions When two functions are inverses, they have a special relationship.
3.4 Use Inverse Funcons •
Inverse Funcons
•
Composites of Inverse Funcons
•
Four steps to finding an inverse funcon
1. They switch domains and ranges. 2. Their composites are equal to x.
Nov 611:11 AM
Nov 611:11 AM
Use Inverse Functions
Use Inverse Functions
When two relations are inverses, they switch their domain and range. Which means they switch their xvalues and y values. Original Relation
x y
2 3
1 4
3 0
6 1
10 8
Nov 611:11 AM
Nov 611:11 AM
Use Inverse Functions
Use Inverse Functions
Original Relation
x y
2 3
1 4
3 0
6 1
10 8
4 1
0 3
1 6
8 10
Inverse Relation
x y
3 2
Nov 611:11 AM
Nov 611:11 AM
1
Algebra 2 Ch 3.4 Use Inverse Functions .notebook
Use Inverse Functions When relations are inverses, they are reflective over the line y=x.
November 27, 2012
Use Inverse Functions If you are given an equation, you can find its inverse.
The inverse of
is
Nov 611:11 AM
Use Inverse Functions
Nov 611:11 AM
Use Inverse Functions Steps for Finding the Inverse of a Function
Find for .
1. Replace 2. Switch
We will use a four step process to find the inverse of a function.
and
3. Solve for 4. Replace
Nov 611:11 AM
Use Inverse Functions 1. Replace
with
with
Nov 611:11 AM
Use Inverse Functions
with 2. Switch
Nov 611:11 AM
and
Nov 611:11 AM
2
Algebra 2 Ch 3.4 Use Inverse Functions .notebook
Use Inverse Functions
November 27, 2012
Use Inverse Functions
3. Solve for 4. Replace
Nov 611:11 AM
Use Inverse Functions
with
Nov 611:11 AM
Use Inverse Functions
Further Simplification of the Problem
Find for
Nov 611:11 AM
Use Inverse Functions 1. Replace
with
Nov 611:11 AM
Nov 611:11 AM
Use Inverse Functions 2. Switch
and
Nov 611:11 AM
3
Algebra 2 Ch 3.4 Use Inverse Functions .notebook
Use Inverse Functions 3. Solve for
Use Inverse Functions 3. Solve for
Nov 611:11 AM
Use Inverse Functions 4. Replace
November 27, 2012
Nov 611:11 AM
Use Inverse Functions
with
Find for
Nov 611:11 AM
Use Inverse Functions 1. Replace
with
Nov 611:11 AM
Nov 611:11 AM
Use Inverse Functions 2. Switch
and
Nov 611:11 AM
4
Algebra 2 Ch 3.4 Use Inverse Functions .notebook
Use Inverse Functions 3. Solve for
Use Inverse Functions 3. Solve for
Nov 611:11 AM
Use Inverse Functions 4. Replace
November 27, 2012
with
Nov 611:11 AM
Use Inverse Functions Example #3 from the book on page 191. An elastic band's resistance R in pounds can be modeled by:
where L is the total length of the stretched rubber band in inches. • Find the inverse of the model. • Use the inverse function to find the length at which the band provides 19 pounds of resistance.
Nov 611:11 AM
Use Inverse Functions 1. Skip step #1
Nov 611:11 AM
Use Inverse Functions 3. Solve for
2. Skip step #2 3. Solve for
Nov 611:11 AM
Nov 611:11 AM
5
Algebra 2 Ch 3.4 Use Inverse Functions .notebook
Use Inverse Functions
November 27, 2012
Use Inverse Functions
4. Skip step #4
Now solve for The rubber band stretches 64 inches when 19 pounds of force is applied to it.
Nov 611:11 AM
Use Inverse Functions
What happens when you make a composite out of inverse functions?
Nov 611:11 AM
Use Inverse Functions
Find:
Nov 611:11 AM
Nov 611:11 AM
Use Inverse Functions The two functions below are inverses, find their composites, f(g(x)) and g(f(x)).
Nov 611:11 AM
Use Inverse Functions
Find:
Nov 611:11 AM
6
Algebra 2 Ch 3.4 Use Inverse Functions .notebook
Use Inverse Functions
November 27, 2012
Use Inverse Functions
The composite of inverses is always equal to x.
Nov 611:11 AM
Use Inverse Functions
Nov 611:11 AM
Use Inverse Functions When two functions are inverses, the domain of the inner function becomes the range of the outer function.
Nov 611:11 AM
Use Inverse Functions
Nov 611:11 AM
Use Inverse Functions
Are and inverses?
You need to find the composites of the functions to prove if they are inverses or not.
Nov 611:11 AM
Nov 611:11 AM
7
Algebra 2 Ch 3.4 Use Inverse Functions .notebook
Use Inverse Functions
November 27, 2012
Use Inverse Functions
They are not inverses, because the composite is not equal to x. You do not have to find the other composite.
Nov 611:11 AM
Use Inverse Functions
Nov 611:11 AM
Use Inverse Functions
If
Find: This just proved what we already knew.
Nov 611:11 AM
Use Inverse Functions
Nov 611:11 AM
Nov 611:11 AM
Use Inverse Functions
Nov 611:11 AM
8
Algebra 2 Ch 3.4 Use Inverse Functions .notebook
Use Inverse Functions Since
Then
November 27, 2012
Use Inverse Functions
Horizontal Line Test is its own inverse.
Nov 611:11 AM
Use Inverse Functions
Nov 611:11 AM
Use Inverse Functions
Nov 611:11 AM
The horizontal line test will tell you if the inverse of a function is itself a function.
Nov 611:11 AM
Use Inverse Functions
Nov 611:11 AM
Use Inverse Functions
Nov 611:11 AM
9
Algebra 2 Ch 3.4 Use Inverse Functions .notebook
November 27, 2012
Use Inverse Functions 3.4 Use Inverse Funcons •
Inverse Funcons
•
Composites of Inverse Funcons
•
Four steps to finding an inverse funcon
Nov 611:11 AM
10
Attachments
Algebra 2 Ch 3.4 Inverse Functions, inverse of itself.gsp
